Chapter 8

Linear Transformations

8.1 Matrix Multiplication as a Linear Transformation

Definition 8.1.1 Let V and W be two finite dimensional vector spaces. A function, Lwhich maps V to W is called a linear transformation and written L ∈ L (V,W ) if for allscalars α and β, and vectors v,w,

L (αv+βw) = αL (v) + βL (w) .

An example of a linear transformation is familiar matrix multiplication. Let A = (aij)be an m× n matrix. Then an example of a linear transformation L : Fn → Fm is given by

(Lv)i ≡n∑

j=1

aijvj .

Here

v ≡

v1...

vn

 ∈ Fn.

8.2 L (V,W ) as a Vector Space

Definition 8.2.1 Given L,M ∈ L (V,W ) define a new element of L (V,W ) , denoted byL+M according to the rule1

(L+M) v ≡ Lv +Mv.

For α a scalar and L ∈ L (V,W ) , define αL ∈ L (V,W ) by

αL (v) ≡ α (Lv) .

You should verify that all the axioms of a vector space hold for L (V,W ) with theabove definitions of vector addition and scalar multiplication. What about the dimensionof L (V,W )?

Before answering this question, here is a useful lemma. It gives a way to define lineartransformations and a way to tell when two of them are equal.

Lemma 8.2.2 Let V and W be vector spaces and suppose {v1, · · · , vn} is a basis for V.Then if L : V →W is given by Lvk = wk ∈W and

L

(n∑

k=1

akvk

)≡

n∑k=1

akLvk =

n∑k=1

akwk

then L is well defined and is in L (V,W ) . Also, if L,M are two linear transformations suchthat Lvk =Mvk for all k, then M = L.

Proof: L is well defined on V because, since {v1, · · · , vn} is a basis, there is exactly oneway to write a given vector of V as a linear combination. Next, observe that L is obviouslylinear from the definition. If L,M are equal on the basis, then if

∑nk=1 akvk is an arbitrary

vector of V,

L

(n∑

k=1

akvk

)=

n∑k=1

akLvk =

n∑k=1

akMvk =M

(n∑

k=1

akvk

)1Note that this is the standard way of defining the sum of two functions.

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