Description
Matrices
1. Evaluate the following matrix-vector products:
a)
2 −4
5 −3
4
−2
b)
1 2 3
−3 −2 −1
1
3
−1
2. Evaluate the following matrix-matrix products:
a)
2 2
3 3 −3 4 2
1 −4 −6
b)
1 1
2 2
3 3
3 −1
4 5
3. For this question, let A be a 1 × 3 matrix and B be a 3 × 1 matrix given by
A =
1 2 3
, B =
1
2
3
.
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.)
Geometrical Transformations
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
0
1
, x0
2
) 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
~vL =
1/
√
2
0
1/
√
2
, ~vA =
1/
√
8
√
3/2
1/
√
8
.
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