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Subspaces of Euclidean Space Rn
Applied Linear AlgebraMATH 5112/6012
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 1 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V.
For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication.
In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V.
For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition.
In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn. (For example, V could be asolution set to some equation, or it could be all the vectors that have thirdcoordinate -7.)
We say that V closed with respect to scalar multiplication if and only ifwhenever ~v is in V and s is any scalar, then s~v is also in V. For example,if V = Span{~v} (for some ~v in Rn), then V is closed with respect toscalar multiplication. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any~v1, ~v2, . . . , ~vp in Rn), then V is closed with respect to scalar multiplication.
We say that V closed with respect to vector addition if and only ifwhenever ~u and ~v are in V, then ~u + ~v is also in V. For example, ifV = Span{~v} (for some ~v in Rn), then V is closed with respect to vectoraddition. In fact, if V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp inRn), then V is closed with respect to vector addition.
We call V a vector subspace of Rn if and only if . . .Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 2 / 12
What is a subspace?
Let V be a collection of vectors in Rn.
We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
What is a subspace?
Let V be a collection of vectors in Rn.We call V a vector subspace of Rn if and only if
~0 is in V,
V closed with respect to vector addition, and
V closed with respect to scalar multiplication.
Some simple examples:
V = {~0} is the trivial vector subspace
V = Rn is a vector subspace of itself (also kinda trivial)
V = Span{~v} (for any ~v in Rn)
V = Span{~v1, ~v2, . . . , ~vp} (for any ~v1, ~v2, . . . , ~vp in Rn)
A simple non-example:
V ={all ~v in R4 with third coordinate -7
}is not a subspace
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 3 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~0
Figure: x + y = 1
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~0
Figure: x + y = 1
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~v
~u
~u + ~v
Figure: xy ≥ 0
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~v
~u
~u + ~v
Figure: xy ≥ 0
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~v
−~v
Figure: x ≥ 0 and y ≥ 0
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
More Examples—Which are, or are not, vector subspaces?
For each V, decide whether or not V is closed with respect to scalarmultiplication and/or closed with respect to vector addition.
V ={[x
y
] ∣∣∣ x + y = 1}
~0 not in V, so not VSS
V ={[x
y
] ∣∣∣ xy ≥ 0}
V not closed wrt vector add
V ={[x
y
] ∣∣∣ x ≥ 0 and y ≥ 0}
V not closed wrt scalar mult
~v
−~v
Figure: x ≥ 0 and y ≥ 0
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 4 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition.
Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means
there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
Let V = Span{~v1, ~v2, . . . , ~vp}. Let’s show that V is closed wrt vectoraddition. Let ~u, ~v be vectors in V. This means there are scalarss1, s2, . . . , sp and t1, t2, . . . , tp with
~u = s1~v1 + · · ·+ sp~vp and ~v = t1~v1 + · · ·+ tp~vp
so
~u + ~v = (s1 + t1)~v1 + (s2 + t2)~v2 + · · ·+ (sp + tp)~vp
which is a vector in V.
Homework: Show that V is closed wrt scalar multiplication.Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 5 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.
Thus is A is an m × n matrix, then CS(A) is a VSS of
Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.
Thus is A is an m × n matrix, then CS(A) is a VSS of
Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.
Thus is A is an m × n matrix, then CS(A) is a VSS of
Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.
Thus is A is an m × n matrix, then CS(A) is a VSS of
Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.Thus is A is an m × n matrix, then CS(A) is a VSS of
Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.Thus is A is an m × n matrix, then CS(A) is a VSS of Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Vector Subspaces—Basic Example
Just saw that any V = Span{~v1, ~v2, . . . , ~vp} is both closed wrt vectoraddition and closed wrt scalar multiplication.
Example (Basic Vector SubSpace)
For any ~v1, ~v2, . . . , ~vp in Rn, Span{~v1, ~v2, . . . , ~vp} is a vector subspace.
In fact, every vector subspace can be expressed this way!
Example (Column Space of a Matrix)
The column space CS(A) of a matrix A is the span of the columns of A.Thus is A is an m × n matrix, then CS(A) is a VSS of Rm.
If A =[~a1 ~a2 . . . ~an
], then CS(A) = Span{~a1, ~a2, . . . , ~an}.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 6 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so,
each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in
Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,
CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Column Space of a Matrix
Let A =[~a1 ~a2 . . . ~an
]be an m × n matrix; so, each ~aj is in Rm.
The column space CS(A) of A is the span of the columns of A, i.e.,CS(A) = Span{~a1, ~a2, . . . , ~an}.
Three Ways to View CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 7 / 12
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b = T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b = T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b = T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0
~b = T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b
= T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b
= T (~p)
Three Ways to View the Column Space CS(A)The column space CS(A) of A =
[~a1 ~a2 . . . ~an
]is:
CS(A) = Span{~a1, ~a2, . . . , ~an}
CS(A) = {~b in Rm∣∣ A~x = ~b has a solution}
CS(A) = Rng(T ) where Rn T−→ Rm is T (~x) = A~x
Rn
~p~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
T (~0) = ~0 ~b = T (~p)
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix.
The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0.
This is avector subspace of
Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0.
This is avector subspace of
Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of
Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0.
This is avector subspace of
Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of
Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
Vector Subspaces—Another Basic Example
Again, let A be an m × n matrix. The null space NS(A) of A is
NS(A) = {~x∣∣ A~x = ~0};
just the solution set for the homogeneous equation A~x = ~0. This is avector subspace of Rn.
Rn
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 9 / 12
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b
= T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
A =[~a1 ~a2 . . . ~an
]an m × n matrix and Rn T−→ Rm is T (~x) = A~x
NS(A) = {~x∣∣ A~x = ~0} and
CS(A) = Span{~a1, ~a2, . . . , ~an}
= {~b in Rm∣∣ A~x = ~b has a solution}
= CS(A) = Rng(T )
Rn
~p
{~x∣∣ A~x = ~b}
NS(A)
~y = T (~x)
~y = A~x
Rm
CS(A) = Rng(T )
~b = T (~p)
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now?
How about row reducing A?
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Null Space and Column Space Example
Find the null space and column space of
A =
1 −1 1 21 0 1 11 2 1 −1
R4
NS(A)
~y = T (~x)
~y = A~x
R3
CS(A) = Rng(T )
What should we do now? How about row reducing A?Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 11 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition
(~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)
If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult
(s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)
If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: then
s1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so
s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
Vector Subspaces—Basic Fact
Recall that a collection V of vectors (in Rn) is a vector subspace (of Rn) ifand only if
~0 is in V,
V closed with respect to vector addition (~u, ~v in V =⇒ ~u + ~v in V)
V closed with respect to scalar mult (s scalar , ~v in V =⇒ s~v in V)If V is a vector subspace; ~v1, ~v2, . . . , ~vp in V; s1, s2, . . . , sp are scalars: thens1~v1, s2~v2, . . . , sp~vp all in V, so s1~v1 + s2~v2 + · · ·+ sp~vp is in V.
Any LC of vectors in a VSS V is a vector in V!
Basic Fact about Vector SubSpaces
Let V be a vector subspace. Suppose ~v1, ~v2, . . . , ~vp are in V.Then Span{~v1, ~v2, . . . , ~vp} lies in V.
Applied Linear Algebra Subspaces of Rn Chapter 3, Section 5 12 / 12
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