Linear Transformations

Def: Linear Transformation

If V and W are vector spaces, a function T: V $\to$ W is called a linear transformation if:

$\forall v_{1}, v_{2} \in V, T (v_{1} \oplus_{V} v_{2}) = T (v_{1}) \oplus_{W} T (v_{2})$
$\forall c \in R, \forall v \in V, T (c ⊙_{V} v) = c ⊙_{W} T (v)$

if $T : V \to V$ then it is often called a linear operator

Proposition: if T is linear then:

T( $0_{V}$ )= $0_{W}$
$\forall v \in V, T (- v) = - T (v)$
If $c_{1}, \dots, c_{n} \in R$ and $v_{1}, \dots, v_{n} \in V$ then: $T (\sum_{i = 1}^{n} c_{i} v_{i}) = \sum_{i = 1}^{n} c_{i} T (v_{i})$

Thm: Linear Extension

Suppose $V$ fdvs and $W$ is a vector space. Let $B = {v_{1}, \dots, v_{n}}$ be a basis for $V$ and for each i=1,...,n let $w_{i} \in W$ . There is a unique linear transformation such that $T (v_{i}) = w_{i}$ .

Def: Kernel

$ker (T) = {v \in V : T (v) = 0_{V}}$

Def: Image

Im( $T$ )= ${T (v) : v \in V}$

Thm: Kernel and Image are subspaces

$T \to W$ is linear then ker(T) and im(T) are subspaces

Thm: Injective if ket(T)= ${0}$

$T : V \to W$ is linear, then T is injective if and only if ker(T)= ${0}$
maps linearly independent vectors to linearly independent vectors
if no basis is in the kernel then it is injective

Thm: im(T) = span(T(basis of pre-image))

Suppose $V$ is a fdvs with basis $B = v_{1}, \dots, v_{n}$ .
If $T : V \to W$ is linear, then im( $T$ )=span( $T (B)$ )

Def: Rank and Nullspace of T

If $T : V \to W$ is linear, then its

rank is rank(T) = dim(im(T))
nullity is nullity(T)=dim(ker(T))

Thm: Rank-Nullity

If $V$ and $W$ are vector spaces and T: $V \to W$ is linear. If rank(T), nullity(T) < $\infty$ then dim(V)=rank(T)+null(T)

pf. define a basis for the image and a map from the pre-image to that basis. Define a basis for the kernel. Show the elements that map to the basis of the image and also the basis of the kernel form a basis for V.

Corollary from Rank-Nullity

If T: V $\to$ W is linear and dim(V)=dim(W) then T is injective $⟺$ T is surjective $⟺$ T is bijective. Leading to the concept of isomorphisms.