Chapter 18

Linear Transformations18.1 Matrix Multiplication As A Linear Transformation

Definition 18.1.1 Let V and W be two finite dimensional vector spaces. A function, Lwhich maps V to W is called a linear transformation and L ∈L (V,W ) if for all scalars α

and β , and vectors v,w,

L(αv+βw) = αL(v)+βL(w) .

These linear transformations are also called homomorphisms. If one of them is one to one,it is called injective and if it is onto, it is called surjective. When a linear transformation isboth one to one and onto, it is called bijective. ,

An example of a linear transformation is familiar matrix multiplication. Let A = (ai j)be an m×n matrix. Then an example of a linear transformation L : Fn 7→ Fm is given by

(Lv)i ≡n

∑j=1

ai jv j.

Here

v≡

v1...

vn

 ∈ Fn.

18.2 The Linear Maps as a Vector SpaceIn what follows I will often continue the practice of denoting vectors in bold face to distin-guish them from scalars. However, this does not mean they are in Fn.

Definition 18.2.1 Given L,M ∈L (V,W ) define a new element of L (V,W ) , denoted byL+M according to the rule

(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 )?

Lemma 18.2.2 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.

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