MAST10007 Linear Algebra Summer Assessment, 2021
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Summer Assessment, 2021
School of Mathematics and Statistics
MAST10007 Linear Algebra
Question 1 (7 marks)
(a) Consider the following flow diagram:
40 x1 60
30 x5 10
At each vertex, the flow in must equal the flow out.
(i) Starting from the top left vertex and reading clockwise, specify four linear equations involving x| , . . . , x5 and write these with all the variables on the left hand side, and constants on the right hand side.
(ii) You are told
┌ 1(1) ' 0 |
_1 0 0 1 |
0 1 0 1 |
0 _1 1 0 |
0 0 1 1 |
|
140 ┐ ' 20 '' . |
Explain why it follows from this that the reduced row echelon form of the augmented matrix corresponding to the linear system in (a) is
3(7)0(0) ┐'
0(10) '' .
(iii) Find the solution of the linear system in (i), given that it is required x3 = 0 and
x5 = 10.
(b) A system of three simultaneous equations in the unknowns x, y, z has only two solutions
(x, y, z) = (0, 0, 0) and (x, y, z) = (1, 0, 0). Is it possible for these equations to be linear? Clearly state the theory being used.
(c) Let k e R be given and consider the system of equations for the unknowns x, y, z specified by
x + ky + 4z = 0
2x _ y + 8z = 0.
Calculate the solution space assuming k _ .
Question 2 (6 marks)
(a) Let E|, E2 be n × n elementary matrices (n > 3) such that for any n × n matrix A
E|A = A │R14R2, E2A = A │R3→R3+2R1 ,
where the notation on the right hand side of each equation indicates a particular elemen- tary row operation applied to A.
(i) For a particular n × n (n > 3) matrix B you are told det B = 6. Calculate det(E|E2B).
(ii) For the matrix B in (i), show that B2 is invertible, clearly stating the theory you are
using.
(iii) Let n = 3. You are told that for a particular 3 × 3 matrix C ,
E|E2C = I3 ,
where I3 denotes the 3 × 3 identity. Specify the entries of C-| .
(b) (i) For what values of d is the matrix
0(0)┐
d'
invertible?
(ii) For the values of d found in (i), use a method based on row operations to compute
D-|.
Question 3 (6 marks)
(a) Consider the following graph
a b
c
e d
and let A be the corresponding adjacency matrix, with vertices a, . . . , e in order corre- sponding to the rows and columns.
(i) Specify A.
(ii) Without doing any actual matrix multiplication, predict the value of the entry in row 4, column 5, of A4 . Explain your reasoning.
(b) (i) For x|, x2 , x3 e R, and given
X = '(┌) '(┐) , Y = '(┌)_1191 2(2)1(0) 4(1)2(1)'(┐) ,
calculate
Tr (XY) and Tr (YX).
(ii) Show that there do not exist 3 × 3 square matrices X|, X2 such that
X|X2 _ X2X| = ┌┐
Clearly state any theory that is being assumed.
Question 4 (6 marks)
Consider the three points in R3
x1 = (_1, 5, 8),
and let
a = x1 _ x2 ,
(a) Find the value of t e R such that b + tc is perpendicular to a. (b) Find the volume of the parallelepiped specified by the vectors x1 , x2 , x3 .
(c) Use your answer to (b) to determine the volume of parallelepiped specified by the vectors x1 + x2 + x3 , x2 + x3 , x3 .
Question 5 (7 marks)
(a) Consider the set
S = {x e R3 : x . (1, 1, 1) = 0}.
(i) Describe S geometrically, and use knowledge of the geometrical form of subspaces in R3 to conclude that it is a subspace.
(ii) Use the subspace theorem to provide an alternative method to show S is a subspace. (b) Show that the set Q = {A e M2|2 : A is singular} is not a subspace of M2|2 .
Question 6 (7 marks)
It is possible to apply elementary row operations to the following matrix A to obtain the matrix
B:
_4
2
2
Using this information, or otherwise, answer the following questions:
(a) Write down a basis for the column space of A, and explain your reasoning.
(b) Do the vectors {(3, _4, 1, 8), (_2, 2, _1, _5), (1, 2, 2, 1)} span a 3-dimensional subspace of
R4 ? Give a reason.
(c) Are the vectors {(3, _2, 1), (_4, 2, 2), (8, _5, 1)} linearly independent? Give a reason.
(d) Write (8, _5, 1) as a linear combination of (3, _2, 1) and (_4, 2, 2).
(e) What is the nullity of A? Give a reason.
(f) Find a basis for the solution space of A.
Question 7 (6 marks)
(a) Let R be the linear transformation R : R2 → R2 specified by an anti-clockwise rotation of π/4 about the origin. Let S be the linear transformation S : R2 → R2 specified by a reflection in the line y = x.
(i) Specify the standard matrix representations of both R and S .
(ii) Specify the standard matrix representation of the linear transformation correspond-
ing to applying R eight times, then applying S twice.
(b) Define the linear transformation Q : p2 → p| by
d
dx
(i) What is the size of the standard matrix [Q]?
(ii) Calculate [Q].
Question 8 (6 marks)
(a) Let T : R3 → R3 be the transformation
T (x1 , x2 , x3 ) = (x1 + x2 , x1 _ x3 , x2 + x3 ).
(i) Find the standard matrix [T] of T. (ii) Find a basis for the kernel of T. (iii) Is T invertible? Give a reason.
(b) Let wˆ = (1, 1, 2).
(i) Give a geometrical interpretation of the linear transformation T : R3 → R3 specified by
T (x) = (x . wˆ )wˆ .
(ii) Specify Im T.
(iii) Is T invertible? Give a reason.
Question 9 (6 marks)
(a) Verify that
B = {(0, 1, 0), (1, 1, 1), (0, 0, 1)}
is a basis for R3 .
(b) Write down the transition matrix pS |B from the basis B to the standard basis s .
(c) Calculate the transition matrix pB |S .
(d) You are given that for a particular linear transformation T : R3 → R3 ,
[T]S = ┌ ┐
'_1 0 2' .
Calculate [T]B .
(e) Calculate [T (1, 1, 1)]B .
Question 10 (6 marks)
A 2 × 2 matrix A is such that it can be written
A = A| + A2 , where A| = 2 ┌ ┐1(1) [1 1] , A2 = _ ┌ 1_1┐ [1 _1] .
(a) Calculate
A ┌ ┐1(1) and A ┌ 1_1┐ .
(b) From your answer to (a), or otherwise, specify the eigenvalues and corresponding nor-
malised eigenvectors of A.
(c) What property of A implies that the eigenvectors are orthogonal using the dot product on R2?
(d) (i) Write A in the form A = QDQT for some diagonal matrix D and orthogonal matrix Q.
(ii) Specify the corresponding decomposition for A2 .
Question 11 (6 marks)
Daily sales of a new brand of iced coffee at a particular convenience store were recorded at the end of day x to be equal to y items as given by the table of data:
x y
1 6
2 5
3 5
(a) Find the line of best fit to the data, using the method of least squares. (b) Draw the line of best fit on a graph, and mark in the data points.
(c) Estimate the number of sales on day 6.
Question 12 (6 marks)
(a) Use the Gram-Schmidt procedure to find an orthonormal basis for the subspace W of R4
spanned by the vectors
{(2, 0, 1, 2), (1, 2, 3, 2), (2, 5, 5, 0)}
using the dot product as inner product on R4 .
(b) Find the closest vector in W to (1, 1, 1, 1).
2023-02-15