(1) Under what conditions on u and v are the two vectors u + v and u ? v orthogonal?
[4 marks]
(2) Let a1, a2, a3 be linearly dependent vectors in R
3
, which are not all zero.
Consider the matrix A with a1, a2, a3 as its column vectors.
What are the possible values of rank A? [4 marks]
(3) Let A and B be n × n invertible matrices.
Explain, with reasons, the truth of the following statements.
(a) True or false? For all invertible matrices A and B, (AB)
?1 = A?1B?1
.
[2 marks]
(b) True or false? For all invertible matrices A and B, (AB)
?1 6 = A?1B?1
.
[2 marks]
(4) Let S ? R
n be a non-empty subspace of R
n
.
TRUE or FALSE: Any subset A ? S which contains the zero vector must also be a
subspace of R
n
. Give a proof or counterexample. [4 marks]
(5) (a) Consider the system of linear equations
3x + 6y + 3z = 9
4x + 8y + 6z = 8
5x + 11y + 8z = 11.
Using Gauss-Jordan elimination, find the solution set of this system. Show
which elementary row operation you are using in each step. [5 marks]
(b) Is the system in part (a) equivalent to the system
x + 2y + z = 3
y + 3z = ?4
2z = ?4?
Explain your reasoning. [3 marks]
(c) Is the system in part (a) equivalent to the system
3x + 6y + 3z = 9
4x + 8y + 6z = 8?
If yes, explain your reasoning. If no, give the solution set of this system.
[2 marks]
(6) (a) Find the intersection between the plane
x + 2y ? z = 3
and the line
x =
?
?
?3
?6
0
?
? + t
?
?
?1
?2
1
?
? .
[4 marks]
(b) Prove that the distance between the two planes n · x = k1 and n · x = k2 is
|k1 ? k2|
||n|| .
[4 marks]
(c) Find the distance between the planes
2x ? 12y + 10z = 16 and ? x + 6y ? 5z = 3.
[2 marks]
(7) Consider 2 × 2 matrices of the form A =
_x005F_x005f_x005F
a b
c d . We can think of 2 × 2 matrices as
vectors with respect to the standard addition and scalar multiplicat
ion operations.
Define the trace of a square matrix as the sum of its diagonal elements. For 2 × 2
matrices the trace is
tr(A) = a + d.
(a) Show the following basic properties of the trace:
(i) tr(A + B) = tr(A) + tr(B),
(ii) tr(kA) = k tr(A),
(iii) tr(A
> ) = tr(A),
(iv) tr(AB) = tr(BA)
for A, B any 2 × 2 matrices and k ∈ R a scalar. [6 marks]
(b) Use the properties in (a) to show that the dot product of matrices defined by
A ? B = tr(A
> B)
satisfies the linearity and symmetry properties of the ordinary dot product of
vectors in R
n
. That is, show the following identities
(i) A ? (B + C) = A ? B + A ? C,
(ii) A ? (kB) = k(A ? B),
(iii) A ? B = B ? A
for A, B any 2 × 2 matrices and k ∈ R a scalar. [3 marks]
(c) Consider an orthogonal 2×2 matrix P. Show that multiplying any two matrices
by an orthogonal matrix leaves their dot product unchanged. That is, verify the
following formula
P A ? P B = A ? B
for any 2 × 2 matrices A, B. [6 marks]
(8) Let M be the matrix
M =
?
?
1 2 0
0 3 0
0 1 1
?
?
(a) Find the eigenvalues of M and the corresponding eigenspaces. [10 marks]
(b) Determine if M is diagonalisable, carefully justifying your answer. If M is
diagonalisable, construct the similarity transformation and resulting diagonal
matrix. [5 marks]
(9) Consider the matrix
Tθ =
?
?
cos(θ) 0 ? sin(θ)
0 1 0
sin(θ) 0 cos(θ)
?
?
(a) Calculate TθT?θ. [1 mark]
(b) Calculate TθTφ. [3 marks]
(c) Let Tθ be the matrix of a linear transformation R
3 → R
3
.
Describe the geometric transformation. [2 marks]
(d) Decide whether Tθ is invertible.
If Tθ is invertible, find its inverse. If Tθ is not invertible, justify your reasoning.
[2 marks]
(10) Let A be an n × n orthogonal matrix.
(a) Prove that det A = ±1. [3 marks]
(b) Hence, determine the determinant of B = (det A)An
. [3 marks]