Math 310, Lesieutre Name:

Quiz #4

September 23, 2015

1. Let T:R2→R2be the linear transformation that rotates points (about the origin) 30◦

counterclockwise.

(a) What is the matrix for T?

−1 0 1

−1

−→

−1 0 1

−1

Remember the strategy to ﬁgure out the matrix for a transformation: you need

to look at the two vectors e1= (1,0) and e2= (0,1), and ﬁgure out what the

transformation does to these two. In this case, a little geometry shows us that

the coordinates for the rotated vectors on the right are

T(e1) = √3

2,1

2!, T (e2) = −1

2,√3

To get the matrix for T, we need to take T(e1) and insert it as the ﬁrst column

of the matrix, and take T(e2) and insert it as the second column:

A="√3

2−1

√3

(b) Is the linear transformation Tone-to-one? Why or why not?

Remember that Tis one-o-one if every bin R2is the image of at most one x.

This is the case for this transformation: given any vector b, there is never more

than one vector that gets sent to b, since the map is just a rotation.

You could also check this using row reduction on the matrix you found in (a):

you need to verify that the columns of Aare linearly independent.

2. Consider the matrices:

A=1−1

0 2 , B =1 3 1

2 2 1.

For each of AB and BA, either compute it or explain why it is not deﬁned.

MATH 310 Quiz: Math 310 Quiz 4

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