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LANCASTER UNIVERSITY
2011 EXAMINATIONS
PART II (Second year)
MATHEMATICS & STATISTICS 2 hours
Math 220: Linear Algebra
You should answer ALL Section A questions and THREE Section B questions.
In Section A there are questions worth a total of 50 marks, but the maximum mark that you can
gain there is capped at 40
SECTION A
A1. Find, in Cartesian form, the equations of the straight line through the point with position
vector (− 1 , 2 , −3) parallel to the direction given by (2, 1 , −1). [5]
A2. Let
A =
(i) Find the inverse of A. [6]
(ii) Hence solve the system of equations
x + y + 2 z = 1
2 x + 4 y − 3 z = 1
3 x + 6 y − 5 z = 1 ,
and CHECK your answer. [4]
A3. Let W = {(x 1 , x 2 , x 3 , x 4 ) : x 2 = 2x 1 + x 3 , x 4 = x 1 − 2 x 3 }.
(i) Prove that W is a subspace of R
4
. [6]
(ii) Find a basis for W. [4]
please turn over
SECTION A continued
A4. Let T : R
2 → R
2 be given by
T ((x, y)) = (x + y, − 3 x + 2y).
(i) Prove that T is a linear transformation. [3]
(ii) Write down the matrix of T with respect to the standard basis in both the domain and
the codomain. [1]
(iii) Write down the transition matrix from the basis (1, 2), (2, 2) to the standard basis in R
2 .
[1]
(iv) Write down the transition matrix from the basis (1, −1), (1, 1) to the standard basis in
R
2
. [1]
(v) Use (ii), (iii) and (iv) to calculate the matrix of T with respect to the basis (1, 2), (2, 2)
in the domain and the basis (1, −1), (1, 1) in the codomain. [4]
A5. By calculating the ranks of two matrices, show that the system of equations below is incon-
sistent:
−x + 2y − 4 z = 7
− 3 x + 2y + 3z = 3
− 8 x + 8y − 2 z = − 7.
[5]
A6. For each v = (x 1 , y 1 ), w = (x 2 , y 2 ) ∈ R
2 define
(v, w) = 2x 1 x 2 + 3y 1 y 2.
(i) Show that this is an inner product on R
2 , and [7]
(ii) find the length of (1, 1) with respect to this inner product. [3]
please turn over
SECTION B continued
B2. (^) (i) Let V, W be finite-dimensional vector spaces over the field F and let T : V → W be a
linear transformation from V to W. Prove that ker T is a subspace of V. [3]
(ii) Let T : R
4 → R
4 be the linear transformation which is represented by the matrix
A =
with respect to the standard basis in both the domain and the codomain.
(a) Find the reduced echelon form of A. [4]
(b) Determine the kernel of T , and write down a basis for it. [3]
(c) Find the dimension of the image of T , justifying your answer. [2]
(d) Determine a basis for im T. [2]
(e) Show that the vector v = (− 1 , 2 , 5 , 8) belongs to im T. [2]
(f) Find the particular solution to the system of equations
x + 2y + 3z + 4w = − 1
2 x + 3y + 4z + 5w = 2
3 x + 4y + 5z + 6w = 5
4 x + 5y + 6z + 7w = 8
with z = w = 0. [2]
(g) Hence write down the general solution to the system of equations in (f). [2]
please turn over
SECTION B continued
B3. (^) (i) Let V be the vector space of all functions with domain [0, 1] and codomain R in which
addition and scalar multiplication of functions is defined by
(f + g)(x) = f (x) + g(x), (λf )(x) = λf (x) for all x ∈ [0, 1], λ ∈ R.
Determine whether each of the following two subsets S 1 , S 2 of V are subspaces of V :
(a) S 1 = {f ∈ V : f (0) = 0};
(b) S 2 is the set of functions f : [0, 1] → R given by
f (x) = ax + 1 for some a ∈ R, for all x ∈ [0, 1].
[6]
(ii) Let R
∞ be the vector space of all infinite sequences (a 1 , a 2 , a 3 ,... , an,.. .) of real numbers
and let T : R
∞ → R
∞ be defined by
T ((a 1 , a 2 , a 3 ,... , an,.. .)) = (a 1 + a 2 , a 2 + a 3 , a 3 + a 4 ,... , an + an+1,.. .),
for all sequences (a 1 , a 2 , a 3 ,... , an,.. .) ∈ R
∞ .
(a) Prove that T is a linear transformation. [4]
(b) Find a basis for ker T and write down the dimension of ker T. [5]
(c) Show that (2, 4 , 8 ,... , 2
n ,.. .) is an eigenvector of T and write down the correspond-
ing eigenvalue. [3]
(d) Write down an eigenvector with eigenvalue 11. [2]
B4. Let
A =
(i) Define the terms eigenvalue, eigenvector and eigenspace for A. Prove that every eigenspace
of an n × n matrix is a subspace of R
n
. [6]
(ii) Determine the characteristic equation of A, and hence find the eigenvalues of A. [4]
(iii) Determine the eigenspaces of A. [6]
(iv) Hence find an orthonormal basis for R
3 consisting of eigenvectors of A. [2]
(v) Write down a matrix P and a diagonal matrix D such that P
T AP = D. [2]
end of exam
