[SOLVED] Math 231 — hw 17

The null space is the space of all vectors that are sent to 0 by a matrix. For example, the
null space of

2 4
−1 −2

is the set of vectors of the form

2x
−x

.
To demonstrate this, we see that

2 4
−1 −2
  2
−1

=

0
0

.
1. Consider the following matrices. What is their null space? Based on their null space,
do their column vectors form a basis?
(a)

3 3
−1 −1

(b)


1 0 1
0 1 1
1 1 2


2. If a null space has more than just the 0 vector, we call it “nontrivial.” Give the basis
of the nontrivial null space of the following matrix

1 −1 2
3 −3 6
3. Suppose M is a 3 × 3 matrix. We said in class that M can be thought of as changing
the basis of the matrix. For this reason, the columns of M represent a basis. If the
null space is nontrivial, then the vectors don’t form a basis of the 3 dimensional vector
space. What does that mean about dimensionality of the range of M?
4. Based on your answer to the previous question, what does it mean geometrically if a
matrix has a nontrivial null space?
5. To capture rotations in two dimensions, we can use the following matrix:

cos θ − sin θ
sin θ cos θ

Suppose a camera is pointed downward looking at a specimen located at (2, 3). If
the camera is rotated by 240 degrees in the positive direction, what is the vector that
represents its location with this new orientation?