Linear Algebra, Theory And Applications, 2012a

Linear Transformations
9.1 Matrix Multiplication As A Linear Transformation
Definition 9.1.1 Let V and W be two finite dimensional vector spaces. A function, L
which maps V to W is called a linear transformation and written L ∈L(V,W) if for all
scalars 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 =(a ij )
be an m × n matrix. Then an example of a linear transformation L : F n → F m is given by
(Lv) i
≡
n∑
a ij v j .
j=1
Here
⎛
⎜
v ≡ ⎝
v 1
. .
v n
⎞
⎟
⎠ ∈ F n .
9.2 L (V,W) As A Vector Space
Definition 9.2.1 Given L, M ∈L(V,W) define a new element of L (V,W) , denoted by
L + M according to the rule 1
(L + M) v ≡ Lv + Mv.
For α ascalarandL ∈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) withthe
above definitions of vector addition and scalar multiplication. What about the dimension
of L (V,W)?
Before answering this question, here is a useful lemma. It gives a way to define linear
transformations and a way to tell when two of them are equal.
1 Note that this is the standard way of defining the sum of two functions.
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