Chapter 5

Linear TransformationsThis chapter is on functions which map a vector space to another one which are also lin-ear. The description of these is in the following definition. Linear algebra is all aboutunderstanding these kinds of mappings.

Definition 5.0.1 Let V and W be two finite dimensional vector spaces. A function, L whichmaps V to W is called a linear transformation and written L ∈L (V,W ) if for all scalarsα and β , and vectors v,w, L(αv+βw) = αL(v)+βL(w) .

Example 5.0.2 Let V =R3,W =R, and let a∈R3 be given vector in V. Define T : V →Wby Tv ≡ ∑

3i=1 aivi

It is left as an exercise to verify that this is indeed linear. Here is an interesting obser-vation.

Proposition 5.0.3 Let L : Fn → Fm be linear. Then there exists a unique m× n matrix Asuch that Lx= Ax for all x. Also, matrix multiplication yields a linear transformation.

Proof: Note that x= ∑ni=1 xiei and so

Lx = L

(n

∑i=1

xiei

)=

n

∑i=1

xiLei =(

Le1 · · · Len

)x1...

xn

=

(Le1 · · · Len

)x

The matrix is A. The last claim follows from the properties of matrix multiplication. ■I will abuse terminology slightly and say that a m×n matrix is one to one if the linear

transformation it determines is one to one, similarly for the term onto.

5.1 L (V,W ) as a Vector SpaceThe linear transformations can be considered as a vector space as described next.

Definition 5.1.1 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 the abovedefinitions of vector addition and scalar multiplication. In fact, is just a subspace of the setof functions mapping V to W which is a vector space thanks to Example 3.0.3. What aboutthe dimension of L (V,W )? What about a basis for 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.

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

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