Review exercise 1.29. Find 2 by 2 matrices that will
(a) reverse the direction of a vector
(b) project a vector onto the 
axis
(c) rotate a vector counter-clockwise through 90 degrees
(d) reflect a vector about the line 
that is 45 degrees above the 
axis
Answer: (a) We want a matrix 
such that 
. We have 
where 
is the 2 by 2 identity matrix, and thus 
. So we have if 
(b) We want a matrix such that 
where 
. By experiment we see that the following matrix will work
(c) We want a matrix such that where 
is a vector rotated 90 degrees counter-clockwise from 
. Such a matrix would send, e.g., the vector 
to 
and in general would send the vector 
to 
. By experiment we see that the matrix
will do this:
(d) We want a matrix such that where is a vector reflected about the line 
. Such a matrix would send, e.g., the vector to 
and in general would send the vector to 
. By experiment we see that the matrix
will do this:
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition
by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition
, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition
and the accompanying free online course, and Dr Strang’s other books
.
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