5) Linear Transformations

Question 1

What is a linear transformation

Answer 1

T : V → W such that for all u, v ∈ V and all λ ∈ K,
(a) T(u + v) = T(u) + T(v);
(b) T(λu) = λT(u).

Question 2

What are some properties of a linear transformation

Answer 2

• T(0) = 0
• T(−v) = −T(v) for all v ∈ V
• T(u − v) = T(u) − T(v) for all u, v ∈ V

Question 3

What is the kernal of a linear transformation

Answer 3

The set of all vectors in V that are mapped to 0 by
T: Ker(T) = {v ∈ V | T(v) = 0} ⊆ V

Question 4

What is the image of a linear transformation

Answer 4

The image of a linear transformation T : V → W is the image of T as a function,
i.e., Im(T) = {w ∈ W|w = T(v) for some v ∈ V } ⊆ W

Question 5

What is Ker(T) and Im(T) a subspace of

Answer 5

Let T : V → W
Ker(T) is a subspace of V
Im(T) is a subspace of W

Question 6

Describe the proof that
* Ker(T) is a subspace of V
* Im(T) is a subspace of W

Answer 6

• (a) Since T(0) = 0, we have 0 ∈ Ker(T) and so Ker(T) ≠∅.
• Let u, v ∈ Ker(T) and λ ∈ K. Then T(u + v) = T(u) + T(v) = 0 + 0 = 0 and so u + v ∈ Ker(T), and T(λu) = λT(u) = λ0 = 0 and so λu ∈ Ker(T).
• By the subspace criteria, Ker(T) is a subspace of V .
• (b) Let x, y ∈ Im(T). So x = T(u), y = T(v) for some u, v ∈ V .
• Then x + y = T(u) + T(v) = T(u + v) and so x + y ∈ Im(T) and if λ ∈ K, then λx = λT(u) = T(λu) and so λx ∈ Im(T).
• Therefore Im(T) is a subspace of W.

Question 7

What is the nullity and rank of a linear transformation

Answer 7

• dim(Ker(T)) is the nullity of T
• dim(Im(T) is the rank of T

Question 8

When is a linear transformation injective

Answer 8

Let T : V → W be a linear transformation. T is injective if and only if Ker(T) = {0}

Question 9

Descrive the proof that a linear transformation is
injective if Ker(T) = {0}

Answer 9

• If T is injective and v ∈ V with v ≠ 0, then T(v) ≠ T(0) = 0 and so v ∉ Ker(T).
• Therefore Ker(T) = {0}.
• Conversely suppose Ker(T) = {0} and let u, v ∈ V with T(u) = T(v). Then 0 = T(u) − T(v) = T(u − v) ⇒ u − v ∈ Ker(T) ⇒ u − v = 0 ⇒ u = v.
• Therefore T is injective.

Question 10

Is the composite of a linear transformation also a linear transformation

Answer 10

If T : U → V and S : V → W are linear transformations, then the
composite S ◦ T : U → W is also a linear transformation.

Question 11

Describe the proof that a composite linear transformation is also a linear transformation

Answer 11

Question 12

Describe the proof that the inverse of a linear transformation is also a linear transformation

Answer 12

Question 13

When is a linear transformation an isomorphism

Answer 13

• A linear transformation T : V → W is called an isomorphism if it is a bijection.
• We say that two vector spaces V and W are isomorphic (V ∼= W) if there exists an isomorphism T : V → W.

Question 14

What isomophorism exists for a field over K with dim(V) = n

Answer 14

Any finite-dimensional vector space V over a field K with dim(V ) = n is isomorphic to K^n

Question 15

What does B [T]C [v] B equal

Answer 15

Question 16

What does B [S ◦ T] D equal

Answer 16

Question 17

What does C [T^−1] B equal

Answer 17

Question 18

If V is a finite-dimensional vector space with bases B and C and T : V → V, then what does [T] C equal

Answer 18

Question 19

What are similar matrices

Answer 19

If matrices A and B can be written as B = P^−1AP for some invertible matrix P,
then we say that A and B are similar matrices.