Chapter 3 - Linear Maps

Question 1

Definition 1.1 : Linear Map

Answer 1

Let V and W be vector spaces over the same field F(bar). A function T:V -> W is called a linear map or a linear operator if
i) T(v+w) = Tv + Tw (v,w ϵ V)
ii) T(αv) = αT(v) (v ϵ V, α ϵ F(bar))

Question 2

Definition 1.6 : Linear Isomorphism

Answer 2

A bijective linear map is called a linear isomorphism. Two vector spaces V and W are said to be linearly isomorphic (denoted V ≅ W), if there exists a linear isomorphism T: V -> W.
Note: If T: V -> W is a linear isomorphism then it can be shown that T: W -> V is also a linear isomorphism.

Question 3

Theorem 1.7 - Linear maps and bases

Answer 3

Let T: V -> W be a linear map and suppose V has a basis B. Let T(B) = {T(b) : b ϵ B}. Then,
i) T is one-to-one if and only if T(B) is a linearly independent set.
ii) T is onto if and only if <T(B)> = W.
iii) T is an isomorphism if and only if T(B) is a basis of W.

Question 4

Definition 2.1 : Kernel, image and rank of a linear map

Answer 4

Let T: V -> W be a linear map.
- The KERNEL of T (denoted ker T) is defined to be the set {v ϵ V : T(v) = 0}
- The IMAGE of T (denoted im T) is defined to be the set T(V) := { w ϵ W: w = T(v) for some v ϵ V}
- The RANK of T (denoted rank T) is the dimension of im T.

Question 5

Theorem 2.3: Dimension theorem

Answer 5

Let V be a finite-dimensional vector space and let T: V -> W be a linear map. Then,
dimV = rankT + dim(ker T)

Question 6

Definition 5.1 : Eigenvalues and Eigenvectors

Answer 6

Let V be a vector space over F(bar), and let T ∈ L(V). We say that λ ∈ F(bar) is an eigenvalue of T if there exists v_λ ∈ V \ {0} such that Tv_λ = λv_λ. The vector v_λ is called an eigenvector associated to λ.
Furthermore, we shall say that λ is an eigenvalue of A ∈ M_n(Fbar) if it is an eigenvalue of the linear map x |→ Ax (x ∈ Fbar^n), i.e., if there exists x ∈ Fbar^n \ {0} such that Ax = λx.
If this last happens, we shall also say that x is an eigenvector of A.

Question 7

Theorem 5.3

Answer 7

Let V be a finite-dimensional vector space over an algebraically closed field Fbar. Then every T ∈ L(V) has at least one eigen value.

Question 8

Theorem 5.5

Answer 8

Let T ∈ L(V), let λ_1,λ_2,…,λ_m be distinct eigenvalues of T and let v_1,v_2,…,v_m be associated eigenvectors(one per each eigenvalue listed m=m). Then v_1,v_2,…,v_m are linearly independent.