Linear Algebra Transformations

Question

The linear transformation T : R2 → R2 given by T (x) =

[3 4

2 3]

x is one-to-one and onto.

Let ~b =

[c

d]

. Find the unique vector x so that T (b) = x. (The entries of x will be expressed in

terms of c and d.)

Answer 1

Let A be the 2x2 matrix [3 4

2 3]

Let b be the column matrix [c

d]

T(b) = A(b) = [3c + 4d

= x

2c + 3d]