Chapter 2

Question 1

Let V and W be vector spaces over a field F. What is linear mapping?

Answer 1

A mapping T: V → W is called a linear mapping if:

1. T(𝐮+𝐮’) = T(𝐮) + T(𝐮’) for all 𝐮, 𝐮’ ⋲ V
2. T(ɑ𝐮) = ɑT(𝐮) for all ɑ ⋲ F and 𝐮 ⋲ V

Question 2

How can the definition of linear mapping prove T(𝟎) = 𝟎?

Answer 2

Substitute in ɑ = 0

Question 3

How to show something is not a linear map?

Answer 3

Find a specific counter example

Question 4

What is the identity map?

Answer 4

Id: V → V

Does nothing

Question 5

What is the zero map?

Answer 5

V → V, 𝐮 → 𝟎

Question 6

Does any matrix define a linear map?

Answer 6

Yes

Question 7

What matrix is the identity map given by?

Answer 7

The 1 by 1 matrix (1)

Question 8

What matrix is the zero map given by?

Answer 8

The 1 by 1 matrix (0)

Question 9

For a linear mapping T: F^n → F^m, what is the matrix form?

Answer 9

𝐮 → A𝐮 for some matrix A

Question 10

For the linear map T ○ S : U → W, which function is applied first?

Answer 10

S

Question 11

What is the kernel?

Answer 11

Let T: V → W be a linear map between vector spaces.

The kernel of T is the subset

kerT = { 𝐮 ⋲ V: T(𝐮) = 𝟎}.

Question 12

What can the kernel also be referred to as?

Answer 12

The nullspace

Question 13

What is the image?

Answer 13

The image of T is the subset

ImT= { T(𝐮): 𝐮 ⋲ V}.

Question 14

For T: V → W, how are kerT and ImT linked to the vector spaces?

Answer 14

• kerT is a subspace of V

* ImT is a subspace of W

Question 15

How to prove a kernel or image is a subspace?

Answer 15

Same process as before - zero element, addition and scalar multiplication

Question 16

For 𝐮 → A𝐮, how to find Im(T)?

Answer 16

ImT = span(columns of A)

Question 17

Let T: V → W be a linear mapping. What is the nullity of T?

Answer 17

The dimension of its kernel

n(T) = dim ker T

Question 18

Let T: V → W be a linear mapping. What is the rank of T?

Answer 18

The dimension of the image

r(T) = dim im T

Question 19

What is the rank-nullity formula?

Answer 19

Let T: V → W be a linear map between finite-dimensional vector spaces. Then

r(T) + n(T) = dim V

Question 20

How to go about proving rank-nullity formula?

Answer 20

• choose basis for ker T and extend it to a basis for V
• S = {𝐮₁, …, 𝐮_k, 𝐯₁, …, 𝐯_r}
• goal is to show that the T(𝐯)s are a basis of Im T
• if they are then dim V = k + r = dim kerT + dim imT

Question 21

For a mapping between two sets f: X → Y, when is it injective?

Answer 21

If f(x) ≠ f(x’) for any x ≠ x’

Question 22

For a mapping between two sets f: X → Y, when is it surjective?

Answer 22

If for all 𝑦 ⋲ Y, there exists a 𝒙 ⋲ X such that 𝑦 = f(𝒙)

Question 23

For a mapping between two sets f: X → Y, when is it bijective?

Answer 23

If it is injective and surjective

Question 24

What is another name for an injective mapping?

Answer 24

1-1

Question 25

What is another name for a surjective mapping?

Answer 25

Onto

Question 26

For T: V → W, how are injectivity and surjectivity linked to image and kernel?

Answer 26

Mapping is 1. Injective iff kerT = {𝟎} 2. Surjective iff imT=W

Question 27

Suppose V and W are vector spaces over F. What is an isomorphism?

Answer 27

T: V → W is an isomorphism if it is a bijective linear map

Question 28

What does an isomorphism do?

Answer 28

Assigns every element in V to a unique element in W, and covers all of W

Question 29

What does it mean if two spaces V and W are isomorphic?

Answer 29

There is an isomorphism between the 2 spaces, V ≅ W

Question 30

Is an n-dimensional vector isomorphic to F^n?

Answer 30

Yes

Question 31

Is any vector field isomorphic to itself?

Answer 31

Yes

Question 32

Given a vector space V over a field F with basis S = {𝐯₁, ..., v_n}, what is the coordinate vector?

Answer 32

The coordinate vector is the unique vector [𝐯]_s = (𝛂₁, 𝛂₂, ..., 𝛂_n) ⋲ F^n where 𝐯 = 𝛂₁𝐯₁ + ... + 𝛂n𝐯n

Question 33

Let T: V → W be a linear map, S = {𝐮₁, ..., 𝐮n} be a basis for V and R = {𝐰₁, ..., 𝐰m} be a basis of W. What does the matrix of T w.r.t. the bases S and R of V and W look like?

Answer 33

Matrix with n columns and m rows jth column is a coordinate vector of T(𝐮j) with respect to R [T(𝐮j)]_R

Question 34

T: V → W is a linear mapping and A is the matrix of T with respect to a basis S of V and basis R of W. What is [T(𝐯)]_R equal to?

Answer 34

[T(𝐯)]_R = A[𝐯]_S for all 𝐯 ⋲ V

Question 35

What is the most important thing to do to write a matrix of a linear map?

Answer 35

Define two bases, one for each side of the map.

Question 36

What is the transition matrix?

Answer 36

Let S = {𝐯₁, ..., 𝐯n} and S' = {𝐯'₁, ..., 𝐯'n} be bases of V. Then [𝐯]_S = P[𝐯]_S' where P is the transition matrix

Question 37

Let T: V → W be a linear map. Let A be the matrix of T w.r.t. bases S of V and R of W. Let A' be the matrix of T w.r.t. bases S' of V and R' of W. What does A' equal i.t.o. Q, A and P?

Answer 37

A' = Q⁻¹AP where P is the transition matrix, writing S' i.t.o. S and Q is the transition matrix, writing R' i.t.o. R

Question 38

When are 2 m x n matrices A and A' equivalent?

Answer 38

If they are related by the invertible matrices P and Q as A' = Q⁻¹AP

Question 39

What does non-singular mean?

Answer 39

Invertible

Question 40

When are 2 n x n matrices similar?

Answer 40

If they are related by the non singular matrix P by A' = P⁻¹AP

Question 41

What is true of the trace and determinant of similar matrices?

Answer 41

They are equal

Question 42

Let T: V → W be a linear map and V, W be of finite dimension. What is canonical form?

Answer 42

When there are bases of V and W where the matrix of T w.r.t. these bases takes the matrix form (I_r, 0, 0, 0) where Ir is the r by r identity matrix and r is the rank of A

Question 43

In canonical form, what does I_r stand for?

Answer 43

The r by r identity matrix and r is the rank of A

Question 44

Is every matrix equivalent to a matrix in canonical form?

Answer 44

Yes