54 CHAPTER 2. LINEAR TRANSFORMATIONS

26. Prove that if A−1 exists and Ax = 0 then x = 0.

27. Let

A =

 1 2 3

2 1 4

1 0 2

 .

Find A−1 if possible. If A−1 does not exist, determine why.

28. Let

A =

 1 0 3

2 3 4

1 0 2

 .

Find A−1 if possible. If A−1 does not exist, determine why.

29. Let

A =

 1 2 3

2 1 4

4 5 10

 .

Find A−1 if possible. If A−1 does not exist, determine why.

30. Let

A =

1 2 0 2

1 1 2 0

2 1 −3 2

1 2 1 2

Find A−1 if possible. If A−1 does not exist, determine why.

2.3 Linear Transformations

By 2.13, if A is an m× n matrix, then for v,u vectors in Fn and a, b scalars,

A

 ∈Fn︷ ︸︸ ︷au+ bv

 = aAu+ bAv ∈ Fm (2.19)

Definition 2.3.1 A function, A : Fn → Fm is called a linear transformation if for allu,v ∈ Fn and a, b scalars, 2.19 holds.

From 2.19, matrix multiplication defines a linear transformation as just defined. Itturns out this is the only type of linear transformation available. Thus if A is a lineartransformation from Fn to Fm, there is always a matrix which produces A. Before showingthis, here is a simple definition.

Definition 2.3.2 A vector, ei ∈ Fn is defined as follows:

ei ≡



0...

1...

0

,