1. Evaluate the following matrix-vector products:
? ? 4
1 2 3
−3 −2 −1
2. Evaluate the following matrix-matrix products:
3 3 ? ? −3 4 2
1 −4 −6
4 5 ?
3. For this question, let A be a 1 × 3 matrix and B be a 3 × 1 matrix given by
1 2 3
, B =
Note, we would normally refer to these as row and column vectors. The point is that a vector is really
just a special case of a matrix.
(a) For the matrices above, compute both AB. What familiar operation does matrix multiplication
reproduce in this case?
(b) The operation from part (a) is also known as the inner product when working with vectors.
Now use the same rules of matrix multiplication to compute BA. (This operation between vectors
is called the outer product and turns out to be useful for describing the moment of inertia of
rotating bodies and for various statistical analyses.)
4. Your friend tells you they found the vector (−1,
2, 0) after applying a rotation of 45◦ about the x-axis
to some input vector, ~v = (x, y, z). In other words, (−1,
2, 0) = Rx,π/4~v. Determine the input vector
~v (assuming your friend applied the rotation correctly).
5. Find a matrix, A, to describe each of the transformations in R
2 given below. In each case, it will
be helpful to work out what the output vector, ~v 0 = (x
) will should look like in terms of the
components of an arbitrary input ~v = (x1, x2) using a sketch.
(a) Scale a vector down to to half its initial length (while preserving the orientation).
(b) Reverse the direction of a vector.
(c) Reflect (flip) a vector across the line x1 = x2.
(d) Project a vector onto the x-axis.
6. In the early evening, you notice a couple of interesting objects in the night sky – the Lineara Nebula
and the Algebrais Constellation. Using the rotational axis of the Earth as the z-axis of a coordinate
system (and two orthogonal directions for x and y) you can label the positions of these objects with
the unit direction vectors
, ~vA =
As time passes, the Earth rotates about what you have defined to be the z-axis but since you move
with the Earth you see the night sky rotating through the negative of the angle the Earth actually
rotates. With this in mind, find the directions of the Lineara Nebula and Algebrais Constellation 6
hours after your initial measurement. Does the angle between your direction vectors change over this
7. Consider the rotation matrix in R
2 given by Rθ =
cos θ − sin θ
sin θ cos θ
(a) Show that the columns of Rθ are orthogonal. (Note, this means the columns form a basis for R
– a useful property).
(b) Show that the magnitude of each column is equal to 1. (Note, this means the columns vectors are
unit vectors – also a useful property for a basis).
(c) We can actually test parts (a) and (b) simultaneously using matrix multiplication but first we
must define the transpose of a matrix. So, given an m × n matrix A, the transpose is denoted
AT and defined by (AT
)ij = (A)ji. (Basically, just flip the matrix across the main (top-left to
bottom-right) diagonal.) Now, show that RT
θ Rθ = I. (Notice, 0’s appearing off the main diagonal
tell us what we found in part (a) – that the columns are orthogonal – and 1’s appearing on the
main diagonal tell us what we found in part (b) – that the columns are unit vectors.
We call a matrix satisfying AT A = I an orthogonal matrix. These come in very handy for finding
bases in real-world problems.
Columnspace and Nullspace
8. Let Q =
4 −8 4
−2 4 −2
3 −6 3
(a) Find a vector equation for the solution set of Q (i.e., the solution to the homogeneous equation
Q~x = ~0). What geometric shape does this vector equation represent?
(b) Let ~x =
, ~y =
, ~z =
be three vectors from R
that aren’t in the solution set.
Compute the matrix-vector products Q~x, Q~y, Q~z. Identify a simple geometric shape that contains
these three output vectors.
(c) Confirm that the vector equation for the shape you found in (b) is equivalent to Col(Q).
9. Let M =
1 1 5
0 1 2
3 −5 −1
(a) Put M in RREF. What is the rank of M? What is the solution set if M represents a set of
(b) Let’s denote the columns of M by ~m1, ~m2, and ~m3. Show that the columns are linearly dependent
by finding the general solution to a1 ~m1 + a2 ~m2 + a3 ~m3 = ~0.
(c) What is the nullspace of M?
(d) What is the columnspace of M?
(e) What is the sum of the dimensions of (c) and (d)? What geometric shapes do (c) and (d) represent?
(f) Provide a basis for the columnspace of M.