Math 324: Linear AlgebraSection 6.1: Linear Transformations

Section 6.2: The Kernel and Range of a Linear Transformation

Mckenzie West

Last Updated: November 21, 2019

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Last Time.

– Linear Transformations

– Properties of Linear Transformations

Today.

– Matrix Transformations

– The Kernel of a Transformation

– The Range of a Transformation

– Rank and Nullity

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Recall.

If V and W are vector spaces, a linear transformation fromV to W is a mapping T : V →W that satisfies

(a) T (~v1 + ~v2) = T (~v1) + T (~v2) for all ~v1, ~v2 ∈ V ,

(b) and T (c~v) = cT (~v) for all scalars c and ~v ∈ V .

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Definition.

Another very important linear transformation is a matrix

transformation, T : Rn → Rm given by T (~x) = A~x where A issome fixed m × n matrix.

Theorem 6.2.

Let A be an m × n matrix. Then the transformation T : Rn → Rm

defined by T (~v) = A~v is a linear transformation.

Exercise 1.

Verify that T (~u + ~v) = T (~u) + T (~v) and T (c~v) = cT (~v) if T is amatrix transformation.

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Exercise 2.

Define the linear transformation T : Rn → Rm by

T (~v) =

1 1 0 01 0 2 01 0 0 1

~v .

(a) What are n and m?

(b) What is T (1, 1, 1, 1)?

(c) What is the preimage of (1, 1, 1)?(Hint: You’re trying to find vectors ~v ∈ Rn such thatT (~v) = (1, 1, 1), what is T (~v)?)

(d) What is the preimage of ~0?

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Definition.

The kernel of the linear transformation T : V →W is thecollection of all vectors ~v in V such that T (~v) = ~0. The notationfor this is,

ker(T ) = {~v ∈ V : T (~v) = ~0}.

Note.

The kernel of a linear transformation is the preimage of ~0.

Exercise 3.

Let T : R3 → R3 be defined by T (x , y , z) = (0, y , z). What isker(T )?

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Definition.

The identity transformation T : V → V is the mapT (~v) = ~v . Sometimes we denote this map by IdV .The zero transformation T : V →W is the map T (~v) = ~0.

Exercise 4.

What is ker(IdV )?What is the kernel of the zero transformation T : V →W ?

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Exercise 5.

Consider the transformation T : R3 → R2 defined by the matrix

A =

[−1 −2 42 3 4

]What is ker(T )? Carefully write out what it means for T (~v) = ~0.Answer this question by finding a basis for the kernel.

The Kernel of a Matrix Transformation.

If T : Rn → Rm is a matrix transformation defined by the m × nmatrix A, then ker(T ) = null(A).

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Exercise 5.

Consider the transformation T : R3 → R2 defined by the matrix

A =

[−1 −2 42 3 4

]What is ker(T )? Carefully write out what it means for T (~v) = ~0.Answer this question by finding a basis for the kernel.

The Kernel of a Matrix Transformation.

If T : Rn → Rm is a matrix transformation defined by the m × nmatrix A, then ker(T ) = null(A).

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Brain Break.

Share your name and how many siblings you have.

I have 1 sister and 1 brother, both younger.

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Theorem 6.3.

The kernel of a linear transformation T : V →W is a subspace ofthe domain, V .

Exercise 6.

Prove Theorem 6.3 via the subspace test.

(a) State why ker(T ) contains ~0.

(b) Show that if ~u and ~v are in ker(T )—meaning T (~u) = ~0 andT (~v) = ~0—then ~u + ~v ∈ ker(T ) too.

(c) Show that if c is a scalar, then c~u is also in ker(T ).

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Recall.

The range of a transformation T : V →W is the collection of allpossible images under the transformation. We write

range(T ) = {T (~v) : ~v ∈ V }.

Theorem 6.4.

The range of a linear transformation T : V →W is a subspace ofthe codomain W .

Proof Idea.

Take two generic elements of the range of T , T (~u) and T (~v).Then by the definition of linear transformation,

T (~u) + T (~v) = T (~u + ~v).

Therefore the sum of the two generic elements is also in the formof range(T ). Similarly, this can be one with scalar multiples.

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The Range of a Matrix Transformation.

If T : Rn → Rm is a matrix transformation defined by the m × nmatrix A, then range(T ) = col(A).

Exercise 7.

Consider the transformation T : R3 → R2 defined by the matrix

A =

[−1 −2 42 3 4

]What is range(T )? Answer this question by finding a basis for therange.

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Definition.

Let T : V →W be a linear transformation. The nullity of T ,denoted nullity(T ), is the dimension of ker(T ).The rank of T , denoted rank(T ), is the dimension of range(T ).

Exercise 8.

Consider the derivative transformation, D : P2 → P1. What is thenullity and rank of D?

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Theorem 6.5.

Let T : V →W be a linear transformation from an n-dimensionalvector space V into a vector space W . Then

dim(V ) = rank(T ) + nullity(T ).

Exercise 9.

Does this match up with the fact that if A is an m× n matrix then

n = rank(A) + nullity(A)?

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Exercise 10.

Find the rank and nullity of the matrix transformationT : R3 → R3 defined by

A =

1 0 −60 1 −40 0 0