MATH10202
Three hours
THE UNIVERSITY OF MANCHESTER
LINEAR ALGEBRA A
5th June 2019
09:45-12:45
Answer ALL SIX questions
Electronic calculators are not permitted.
1 of 5
c 2019 The University of Manchester, all rights reserved.
MATH10202
SECTION A
Answer ALL THREE questions in this section.
A1.
Consider the real matrices A =
1 2 0 2 3 6
0 6 0
and B =
1 12 3
0 4
.
(a) Calculate BBT 2A.
(b) Use cofactor expansion to compute the determinant ofA. IsA invertible Explain your answer.
(c) Is the matrix A symmetric or skew symmetric If so, which
(d) Define what it means for a square matrix M to be upper triangular.
(e) Prove that if M and N are n× n upper triangular matrices then the sum M +N and product
MN are both upper triangular. (You should explain clearly the reasoning behind each step
of your proof.)
[15 marks]
A2.
(a) Give an example of two different 2 × 2 row echelon matrices which are row equivalent. Can
two different reduced row echelon matrices ever be row equivalent
(b) If two systems of linear equations over a field have augmented matrices which are row equivalent,
what can be said about their solution sets
(c) Use Gaussian elimination to describe the set of all solutions to the following system of linear
equations over R:
x + 3y + 4z = 13
x 3y 3z = 10
2x + 6y + 9z = 29
Express your answer using one or more parameters if necessary.
(d) Find the inverse of the matrix 0 1 01 3 4
2 3 6
.
(You may use any method, but to be eligible for partial credit for an incorrect answer you
should explain your method and show your working.)
(e) Write down the 2 × 2 elementary matrix corresponding to the row operation r2 → r2 2r1.
Write down the inverse of this matrix and the row operation corresponding to the inverse.
Explain briefly the relationship between this row operation and the original one.
[15 marks]
2 of 5 P.T.O.
MATH10202
A3.
Let u = (u1, u2) and v = (v1, v2) be vectors in R2.
(a) State the definitions of the inner product 〈u | v〉 and the norm ||u||. State what it means for
the two vectors to be orthogonal, and for the vector v to be a unit vector.
(b) Suppose now that {u, v} is an orthonormal basis for R2. What can we say about 〈u | v〉 and
||v|| What can we say about span({u, v})
(c) Still assuming that {u, v} is an orthonormal basis for R2, suppose w is another vector in R2.
Prove that
w = 〈u | w〉u + 〈v | w〉v.
(You may assume basic properties of vector addition, scaling and the inner product. You should
explain your reasoning clearly.)
(d) Consider the set
S = {(1, 1, 1), ( 1, 0, 5), (0, 12, 6)} R3.
You may assume that S is linearly independent, and therefore is a basis for R3. Use the
Gram-Schmidt process to transform S into an orthonormal basis for R3.
[15 marks]
3 of 5 P.T.O.
MATH10202
SECTION B
Answer ALL THREE questions in this section
B4.
(a) Let V be a vector space over a field K, and let B = {v1, . . . , vn} be a subset of V . Define what
is meant by the following:
(i) B is linearly independent;
(ii) B spans V ;
(iii) B is a basis for V .
(b) Let W = {A ∈M3(R)|AT = A}.
(i) Use the subspace test to show that W is a subspace of M3(R). You may use elementary
properties of the transpose of a matrix.
(ii) Write down a basis for W .
(c) Consider the bases B =
