Page 174 - A Course in Linear Algebra with Applications

158 Chapter Six: Linear Transformationns

Exercises 6.1

1. Label each of the following functions F : R —• R injective,
surjective or bijective, as is most appropriate. (You may wish
to draw the graph of the function in some cases):
2 3 2
(a) F(x) = x ; (b) F(x) = x /(x + 1);
(c) F(x) = x(x ~l)(x-2); (d) F(x) = e x + 2.
2. Let functions F and G from R to R be defined by F{x) =
2x — 3, and G{x) = (x 2 — l)/(x 2 + l). Show that the composite
functions F o G and G o F are different.
3. Verify that the following functions from R to R are mutu-
ally inverse: F(x) = 3x — 5 and G(x) = (x + 5)/3.
4. Find the inverse of the bijective function F : R —> R
defined by F(x) = 2x 3 — 5.
5. Let G : F -^ X be an injective function. Construct a
function F : X —• V such that F o G is the identity function
on Y. Then use this result to show that there exist functions
F, G : R -> R such that F o G = 1 R but G o F ^ 1 R .
6. Prove 6.1.2.

7. Complete the proof of part (b) of 6.1.5.

6.2 Linear Transformations and Matrices

After the preliminaries on functions, we proceed at once
to the fundamental definition of the chapter, that of a linear
transformation. Let V and W be two vector spaces over the
same field of scalars F. A linear transformation (or linear
mapping) from V to W is a function

T: V ^ W

with the properties

T(vi + v 2 ) = T(vi) + T(v 2 ) and T(cv) = cT(v)