introduction to linear transformation
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Introduction to Linear Transformation
Math 4A – Xianzhe Dai
UCSB
April 14 2014
Based on the 2013 Millett and Scharlemann Lectures
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System of Linear Equations:
a11x1 + a12x2 + . . .+ a1nxn = b1
a21x1 + a22x2 + . . .+ a2nxn = b2...
am1x1 + am2x2 + . . .+ amnxn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .
Two very nice properties it enjoys are
T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)
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System of Linear Equations:
a11x1 + a12x2 + . . .+ a1nxn = b1
a21x1 + a22x2 + . . .+ a2nxn = b2...
am1x1 + am2x2 + . . .+ amnxn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .
Two very nice properties it enjoys are
T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)
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System of Linear Equations:
a11x1 + a12x2 + . . .+ a1nxn = b1
a21x1 + a22x2 + . . .+ a2nxn = b2...
am1x1 + am2x2 + . . .+ amnxn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .
Two very nice properties it enjoys are
T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)
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System of Linear Equations:
a11x1 + a12x2 + . . .+ a1nxn = b1
a21x1 + a22x2 + . . .+ a2nxn = b2...
am1x1 + am2x2 + . . .+ amnxn = bm
can be written as matrix equation A~x = ~b.
New perspective: think of the LHS as a“function/map/transformation”, T (~x) = A~x . T maps/transformsa vector ~x to another vector A~x .
Two very nice properties it enjoys are
T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)T (c~u) = A(c~u) = cA~u = cT (~u)
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Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,
T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
So T is a linear transformation.
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Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
Example: Let T : R1 → R1 be defined by T (x) = 5x .
For any u, v ∈ R1,
T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
So T is a linear transformation.
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Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,
T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
So T is a linear transformation.
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Linear Transformations
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
Example: Let T : R1 → R1 be defined by T (x) = 5x .For any u, v ∈ R1,
T (u + v) = 5(u + v) = 5u + 5v = T (u) + T (v) and
for any c ∈ R, T (cu) = 5cu = c5u = cT (u).
So T is a linear transformation.
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Some notes:
Most functions are not linear transformations.
For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).
For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then
T (~0) = T (0 ·~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v) = aT (~u) + bT (~v)
.
Example
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Some notes:
Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).
For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then
T (~0) = T (0 ·~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v) = aT (~u) + bT (~v)
.
Example
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Some notes:
Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).
For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5):
Take c = 0, then
T (~0) = T (0 ·~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v) = aT (~u) + bT (~v)
.
Example
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Some notes:
Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).
For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then
T (~0) = T (0 ·~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v) = aT (~u) + bT (~v)
.
Example
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Some notes:
Most functions are not linear transformations. For example:cos(x + y) 6= cos(x) + cos(y). Or (2x)2 6= 2(x2).
For any linear transformation T (~0) = ~0 (this rules outfunction f (x) = x + 5): Take c = 0, then
T (~0) = T (0 ·~0) = 0T (~0) = ~0.
The two conditions could be written as one: For any vectors~u, ~v ∈ Rn and real numbers a, b ∈ R,
T (a~u + b~v) = aT (~u) + bT (~v)
.
Example
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Important example:
Let A be any m × n matrix. Define T : Rn → Rm by T (~x) = A~x .We have already seen that T has what it takes:
For any vectors ~u, ~v ∈ Rn,T (~u + ~v) = A(~u + ~v) = A~u + A~v = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R,T (c~u) = A(c~u) = c(A~u) = cT (~u).
A linear transformation defined by a matrix is called a matrixtransformation.
Important Fact Conversely any linear transformation is associatedto a matrix transformation (by using bases).
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Mona Lisa transformed6/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix transformations are important and are also cool!
Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix
A =
[1 3
20 1
]
For any horizontal vector ~x =
[x10
]
T (~x) = A~x =
[1 3
20 1
] [x10
]=
[x1 + 3
2 · 00 · x1 + 1 · 0
]=
[x10
]So T is the identity on horizontal vectors.
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Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix
A =
[1 3
20 1
]
For any horizontal vector ~x =
[x10
]
T (~x) = A~x =
[1 3
20 1
] [x10
]=
[x1 + 3
2 · 00 · x1 + 1 · 0
]=
[x10
]So T is the identity on horizontal vectors.
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix
A =
[1 3
20 1
]
For any horizontal vector ~x =
[x10
]
T (~x) = A~x =
[1 3
20 1
] [x10
]=
[x1 + 3
2 · 00 · x1 + 1 · 0
]=
[x10
]
So T is the identity on horizontal vectors.
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix transformations are important and are also cool!Example 1, a shear: Consider the matrix transformationT : R2 → R2 given by the 2× 2 matrix
A =
[1 3
20 1
]
For any horizontal vector ~x =
[x10
]
T (~x) = A~x =
[1 3
20 1
] [x10
]=
[x1 + 3
2 · 00 · x1 + 1 · 0
]=
[x10
]So T is the identity on horizontal vectors.
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For any vertical vector ~x =
[0x2
]
T (~x) = A~x =
[1 3
20 1
] [0x2
]=
[1 · 0 + 3
2 · x20 · 0 + 1 · x2
]=
[0x2
]+
[32x20
]So a vertical vector is pushed perfectly horizontally, a distance 3
2times its length:
(0, 1)
(2, 0)
(2, 1)(3/2, 1) (2+3/2, 1)x
1
2
x
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For any vertical vector ~x =
[0x2
]
T (~x) = A~x =
[1 3
20 1
] [0x2
]=
[1 · 0 + 3
2 · x20 · 0 + 1 · x2
]=
[0x2
]+
[32x20
]So a vertical vector is pushed perfectly horizontally, a distance 3
2times its length:
(0, 1)
(2, 0)
(2, 1)(3/2, 1) (2+3/2, 1)x
1
2
x
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For any vertical vector ~x =
[0x2
]
T (~x) = A~x =
[1 3
20 1
] [0x2
]=
[1 · 0 + 3
2 · x20 · 0 + 1 · x2
]=
[0x2
]+
[32x20
]
So a vertical vector is pushed perfectly horizontally, a distance 32
times its length:
(0, 1)
(2, 0)
(2, 1)(3/2, 1) (2+3/2, 1)x
1
2
x
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For any vertical vector ~x =
[0x2
]
T (~x) = A~x =
[1 3
20 1
] [0x2
]=
[1 · 0 + 3
2 · x20 · 0 + 1 · x2
]=
[0x2
]+
[32x20
]So a vertical vector is pushed perfectly horizontally, a distance 3
2times its length:
(0, 1)
(2, 0)
(2, 1)(3/2, 1) (2+3/2, 1)x
1
2
x
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For any vertical vector ~x =
[0x2
]
T (~x) = A~x =
[1 3
20 1
] [0x2
]=
[1 · 0 + 3
2 · x20 · 0 + 1 · x2
]=
[0x2
]+
[32x20
]So a vertical vector is pushed perfectly horizontally, a distance 3
2times its length:
(0, 1)
(2, 0)
(2, 1)(3/2, 1) (2+3/2, 1)x
1
2
x
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Example 2, scaling:Use
A =
[2 00 1
2
]
For any vector ~x =
[x1x2
]T (~x) = A~x =
[2 00 1
2
] [x1x2
]=
[2x1 + 0 · x2
0 · x1 + 12 · x2
]=
[2x1x22
]So T stretches horizontally and contracts vertically:
(0, 1)
(2, 0)
x
1
2
x
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Example 2, scaling:Use
A =
[2 00 1
2
]For any vector ~x =
[x1x2
]T (~x) = A~x =
[2 00 1
2
] [x1x2
]=
[2x1 + 0 · x2
0 · x1 + 12 · x2
]=
[2x1x22
]
So T stretches horizontally and contracts vertically:
(0, 1)
(2, 0)
x
1
2
x
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Example 2, scaling:Use
A =
[2 00 1
2
]For any vector ~x =
[x1x2
]T (~x) = A~x =
[2 00 1
2
] [x1x2
]=
[2x1 + 0 · x2
0 · x1 + 12 · x2
]=
[2x1x22
]So T stretches horizontally and contracts vertically:
(0, 1)
(2, 0)
x
1
2
x
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 2, scaling:Use
A =
[2 00 1
2
]For any vector ~x =
[x1x2
]T (~x) = A~x =
[2 00 1
2
] [x1x2
]=
[2x1 + 0 · x2
0 · x1 + 12 · x2
]=
[2x1x22
]So T stretches horizontally and contracts vertically:
(0, 1)
(2, 0)
x
1
2
x
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Example 3, reflection through a line:Use
A =
[0 11 0
]
T (
[x1x2
]) = A
[x1x2
]=
[0 11 0
] [x1x2
]=
[0x1 + 1 · x2
1 · x1 + 0 · x2
]=
[x2x1
]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:
(s, s)x
1
2
x
reflect
reflect
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Example 3, reflection through a line:Use
A =
[0 11 0
]T (
[x1x2
]) = A
[x1x2
]=
[0 11 0
] [x1x2
]=
[0x1 + 1 · x2
1 · x1 + 0 · x2
]=
[x2x1
]
So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:
(s, s)x
1
2
x
reflect
reflect
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Example 3, reflection through a line:Use
A =
[0 11 0
]T (
[x1x2
]) = A
[x1x2
]=
[0 11 0
] [x1x2
]=
[0x1 + 1 · x2
1 · x1 + 0 · x2
]=
[x2x1
]So T exchanges the two coordinates.
Looks like reflection throughthe line x1 = x2:
(s, s)x
1
2
x
reflect
reflect
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 3, reflection through a line:Use
A =
[0 11 0
]T (
[x1x2
]) = A
[x1x2
]=
[0 11 0
] [x1x2
]=
[0x1 + 1 · x2
1 · x1 + 0 · x2
]=
[x2x1
]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:
(s, s)x
1
2
x
reflect
reflect
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 3, reflection through a line:Use
A =
[0 11 0
]T (
[x1x2
]) = A
[x1x2
]=
[0 11 0
] [x1x2
]=
[0x1 + 1 · x2
1 · x1 + 0 · x2
]=
[x2x1
]So T exchanges the two coordinates. Looks like reflection throughthe line x1 = x2:
(s, s)x
1
2
x
reflect
reflect
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Example 4, rotation:Use
A =
[cos θ − sin θsin θ cos θ
]
T (
[10
]) = A
[10
]=
[cos θ − sin θsin θ cos θ
] [10
]=
[cos θsin θ
]So horizontal unit vector is rotated c-clockwise an angle θ.
Similarly, for the vertical unit vector
[01
], so all of plane rotates:
v
R (v)
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 4, rotation:Use
A =
[cos θ − sin θsin θ cos θ
]T (
[10
]) = A
[10
]=
[cos θ − sin θsin θ cos θ
] [10
]=
[cos θsin θ
]
So horizontal unit vector is rotated c-clockwise an angle θ.
Similarly, for the vertical unit vector
[01
], so all of plane rotates:
v
R (v)
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 4, rotation:Use
A =
[cos θ − sin θsin θ cos θ
]T (
[10
]) = A
[10
]=
[cos θ − sin θsin θ cos θ
] [10
]=
[cos θsin θ
]So horizontal unit vector is rotated c-clockwise an angle θ.
Similarly, for the vertical unit vector
[01
], so all of plane rotates:
v
R (v)
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 4, rotation:Use
A =
[cos θ − sin θsin θ cos θ
]T (
[10
]) = A
[10
]=
[cos θ − sin θsin θ cos θ
] [10
]=
[cos θsin θ
]So horizontal unit vector is rotated c-clockwise an angle θ.
Similarly, for the vertical unit vector
[01
],
so all of plane rotates:
v
R (v)
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Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Example 4, rotation:Use
A =
[cos θ − sin θsin θ cos θ
]T (
[10
]) = A
[10
]=
[cos θ − sin θsin θ cos θ
] [10
]=
[cos θsin θ
]So horizontal unit vector is rotated c-clockwise an angle θ.
Similarly, for the vertical unit vector
[01
], so all of plane rotates:
v
R (v)
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Some types of problems that can come up:
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
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Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.
Find a vector ~u so thatT (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.
Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Some types of problems that can come up:Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~u ∈ Rn is given. What is T (~u)?
Answer: Just do matrix-vector multiplication A~u. The result is avector in Rm.
Question: Suppose T : Rn → Rm is a matrix linear transformation.Suppose A is the matrix of T and ~b ∈ Rm is given.Find a vector ~u so that
T (~u) = ~b.
Answer: Solve the system of equations given by A~x = ~b. Any
solution is such a vector ~u.Reminder: There may be
no solution orexactly one solution ora parameterized family of solutions
12/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
iClicker question
Suppose T is a matrix linear transformation with matrix A below,and we are seeking all vectors ~u so that T (~u) = ~b.
A =
1 2 30 4 50 0 0
~b =
678
How many solutions are there?
A) Zero.
B) One
C) Infinity
D) Can’t tell
What if ~b =
670
?
13/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
iClicker question
Suppose T is a matrix linear transformation with matrix A below,and we are seeking all vectors ~u so that T (~u) = ~b.
A =
1 2 30 4 50 0 0
~b =
678
How many solutions are there?
A) Zero.
B) One
C) Infinity
D) Can’t tell
What if ~b =
670
?
13/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ uniquesolution.
There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy todescribe them.
14/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ uniquesolution.
There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy todescribe them.
14/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ uniquesolution.
There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy todescribe them.
14/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ uniquesolution.
There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy todescribe them.
14/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Question: Describe all vectors ~u so that T (~u) = ~b.
Answer: This is the same as finding all vectors ~u so that A~u = ~b.Could be no ~u, could be exactly one ~u, or could be a parametrizedfamily of such ~u’s.
Recall the idea: row reduce the augmented matrix [A : ~b] to merelyechelon form.
Augmentation column is pivot column ⇐⇒ no solutions.
Augmentation column is only non-pivot column ⇐⇒ uniquesolution.
There are free variables ⇐⇒ there is a parameterized familyof solutions whose dimension is the number of free variables.
If there are solutions, reduced echelon form makes it easy todescribe them.
14/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)
This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .
Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .
15/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
For any linear transformation T (~0) = ~0
T (a~u + b~v) = aT (~u) + bT (~v)
This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .
Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .
15/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)
This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .
Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .
15/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)
This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .
Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .
15/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Definition
A linear transformation is a function T : Rn → Rm with theseproperties:
For any vectors ~u, ~v ∈ Rn, T (~u + ~v) = T (~u) + T (~v)
For any vector ~u ∈ Rn and any c ∈ R, T (c~u) = cT (~u).
For any linear transformation T (~0) = ~0T (a~u + b~v) = aT (~u) + bT (~v)
This has important implications: if you know T (~u) and T (~v) ,then you know the values of T on all the linear combinations of ~uand ~v .
Matrix transformation: Let A be any m × n matrix. DefineT : Rn → Rm by T (~x) = A~x .
15/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
T (
[10
]) =
[52
]; T (
[01
]) =
[34
]
What is T (
[−17
])?
T (
[−17
]) = T (−1
[10
]+ 7
[01
]) = −1T (
[10
]) + 7T (
[01
])
= −1
[52
]+ 7
[34
]=
[1626
]In fact, nothing can stop us from using the same idea to compute
T (
[2−4
]) or T (~x) for any vector ~x ∈ R2:
Example
16/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
T (
[10
]) =
[52
]; T (
[01
]) =
[34
]
What is T (
[−17
])?
T (
[−17
]) = T (−1
[10
]+ 7
[01
]) = −1T (
[10
]) + 7T (
[01
])
= −1
[52
]+ 7
[34
]=
[1626
]In fact, nothing can stop us from using the same idea to compute
T (
[2−4
]) or T (~x) for any vector ~x ∈ R2:
Example
16/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
T (
[10
]) =
[52
]; T (
[01
]) =
[34
]
What is T (
[−17
])?
T (
[−17
]) = T (−1
[10
]+ 7
[01
]) = −1T (
[10
]) + 7T (
[01
])
= −1
[52
]+ 7
[34
]=
[1626
]
In fact, nothing can stop us from using the same idea to compute
T (
[2−4
]) or T (~x) for any vector ~x ∈ R2:
Example
16/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Matrix Is Everywhere
Example: Suppose T : R2 → R2 is a linear transformation so that
T (
[10
]) =
[52
]; T (
[01
]) =
[34
]
What is T (
[−17
])?
T (
[−17
]) = T (−1
[10
]+ 7
[01
]) = −1T (
[10
]) + 7T (
[01
])
= −1
[52
]+ 7
[34
]=
[1626
]In fact, nothing can stop us from using the same idea to compute
T (
[2−4
]) or T (~x) for any vector ~x ∈ R2:
Example16/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
We can carry this much further: All linear transformationsT : Rn → Rm are matrix linear transformations
Why?
~x =
x1x2...xn
= x1
10...0
+ x2
01...0
+ . . . xn
00...1
is a linear combination of the vectors
10...0
,
01...0
, . . . ,
00...1
(standard basis of Rn)
17/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
We can carry this much further: All linear transformationsT : Rn → Rm are matrix linear transformations
Why?
~x =
x1x2...xn
= x1
10...0
+ x2
01...0
+ . . . xn
00...1
is a linear combination of the vectors
10...0
,
01...0
, . . . ,
00...1
(standard basis of Rn)
17/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
So by the property of linear transformation
T (~x) = x1T
10...0
+ x2T
01...0
+ . . . xnT
00...1
only need to know each T (~ej) where
~ej =
0...1...0
← j thentry
Denote ~aj = T (~ej)
T (~x) = x1 ~a1 + x2 ~a2 + . . . xn ~an
18/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
So by the property of linear transformation
T (~x) = x1T
10...0
+ x2T
01...0
+ . . . xnT
00...1
only need to know each T (~ej) where
~ej =
0...1...0
← j thentry Denote ~aj = T (~ej)
T (~x) = x1 ~a1 + x2 ~a2 + . . . xn ~an
18/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Ahha: In matrix notation this is written:
T (~x) =[~a1 ~a2 . . . ~an
]x1x2...xn
= A~x
That is, the matrix
A =[~a1 ~a2 . . . ~an
]is the matrix of T !
19/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Ahha: In matrix notation this is written:
T (~x) =[~a1 ~a2 . . . ~an
]x1x2...xn
= A~x
That is, the matrix
A =[~a1 ~a2 . . . ~an
]is the matrix of T !
19/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Recap:
Since T is a linear transformation,
T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =
x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an
]
x1x2...xn
= A~x .
Example
20/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =
x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an
]
x1x2...xn
= A~x .
Example
20/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =
x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an
]
x1x2...xn
= A~x .
Example
20/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Recap: Since T is a linear transformation,
T (~x) = T (x1~e1+x2~e2+. . . xn~en) = x1T (~e1)+x2T (~e2)+. . . xnT (~en) =
x1~a1 + x2~a2 + . . . xn~an =[~a1 ~a2 . . . ~an
]
x1x2...xn
= A~x .
Example
20/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.
The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.
Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is 1 to 1 if ~u 6= ~v implies that T (~u) 6= T (~v).
Put another words, T (~u) = T (~v) implies ~u = ~v .
If T (~u) = T (~v) we see that T (~u − ~v) = ~0 (T lineartransformation).
Saying that T is 1 to 1 is the same as saying thatT (~w) = ~0 exactly when ~w = ~0 (only trivial solution).
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows.The only solution to the homogeneous equation is the zerosolution. And, as a consequence, n ≤ m.Examples: shears, contractions and expansions, rotations,reflections
21/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .
Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections
22/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .
Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections
22/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .
Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.
The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections
22/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .
Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.
Examples: shears, contractions and expansions, rotations,reflections
22/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rm is a (matrix) linear transformation.
Definition
T is onto if, for any ~v ∈ Rm, is there a ~u ∈ Rnsuch thatT (~u) = ~v .
Saying that T is onto is the same as saying thatT (~u) = ~v always has a solution.
This means that the reduced echelon form of the matrix of T musthave exactly m non-zero rows.The non-homogeneous equation must always have a solution. And,as a consequence, m ≤ n.Examples: shears, contractions and expansions, rotations,reflections
22/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly onesolution.
Examples: shears, contractions and expansions, rotations,reflections
23/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly onesolution.
Examples: shears, contractions and expansions, rotations,reflections
23/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly onesolution.
Examples: shears, contractions and expansions, rotations,reflections
23/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly onesolution.
Examples: shears, contractions and expansions, rotations,reflections
23/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Suppose T : Rn → Rn is a (matrix) linear transformation.
Definition
T is an isomorphism if T is both 1 to 1 and onto.
Saying that T is isomorphism is the same as saying thatT is a bijection that respects the vector space structure.
This means that the reduced echelon form of the matrix of T musthave exactly n non-zero rows, the same as the number of columns.
The non-homogeneous equation must always have exactly onesolution.
Examples: shears, contractions and expansions, rotations,reflections
23/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.
Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.
Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.
Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
Prelude Linear Transformations Pictorial examples Matrix Is Everywhere
Summarizing Suppose T : Rn → Rm is a (matrix) lineartransformation.
T is 1 to 1 if there is a pivot 1 in every column of the reducedechelon form, i.e. there are no free variables.Said differently, the column vectors of the matrix of T are linearlyindependent.
T is onto if there is a pivot 1 in every row of the reduced echelonform.Said differently, the column vectors of the matrix of T span thewhole space Rm.
T is an isomorphism if there is a pivot 1 in every row and column,i.e. the reduced echelon matrix is the identity matrix.Said differently, the column vectors of the matrix of T are linearlyindependent and span the whole space.
24/24
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