Problem 1. The linear transformation T : R2 ) R2 is pictured below. The starting box is a square with sides :1 and r = : The image is a parallelogram...

Find the matrix of

T

in two di erent ways (and check that you get the same result).

(i) Find the images of

e

1

and

e

2

directly from the diagram.

(ii) Write

T

as the composition of, rst a shear

S

, then a dilation

D

, then a re ection

R

; nd the matrix of each, and multiply them together. That is need to nd

Y

of

the form

T

=

RDS

.

(b) Let

Q

be the transformation that applies

T

twice, that is, it maps

x

to

T

(

T

(

x

)). Find

the matrix of

Q

. Check your answer by using the diagram to track the path of the top

left-hand corner of the starting box, and compare with what you get by applying the

matrix to

1

1

.

(c) Find the matrix of the transformation

Z

that inverts"

T

, that is, that maps the paral-

lelogram on the right to the square on the left. Again, check your answer by following a

suitable point.