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LANCASTER UNIVERSITY
2009 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. Show that the vectors (1, 1 , −1), (0, 1 , 3) and (1, − 1 , 1) are linearly independent, and express
(3, 2 , 2) as a linear combination of them. [5]
A2. Let
A =
(i) Find the inverse of A. [6]
(ii) Hence solve the system of equations
y + z = 2
x + y = 1
2 x + 3 y + 2 z = 2,
and CHECK your answer. [4]
A3. Let W = {(x 1 , x 2 , x 3 , x 4 ) : x 3 = x 1 − 2 x 2 , x 4 = x 2 + 3x 1 }.
(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)) = (4x − 5 y, 3 x + 4y).
(i) Prove that T is a linear transformation. [3]
(ii) Find the matrix of T with respect to the basis (1, −1), (1, 1) in both the domain and the
codomain. [3]
(iii) Find the inverse of T. [4]
A5. By calculating the ranks of two matrices, show that the system of equations below is incon-
sistent.
2 x − y + 3 z = 1
x + 2 y − z = − 3
3 x − 4 y + 7 z = 2.
[5]
A6. Let
A =
Given that A has eigenvalues 0, 1 , 6 with corresponding eigenspaces E(0) = sp{(1, − 2 , 1)},
E(1) = sp{(2, 1 , 0)}, E(6) = sp{(− 1 , 2 , 5)}, write down an orthogonal matrix P and a
diagonal matrix D such that D = P
T AP. [5]
A7. Let P be the Euclidean space of all polynomial functions with real coefficients and with inner
product defined by 〈p, q〉 =
0 p(x)q(x)dx.
(i) Find the length of p(x) = 2x + 1. [2]
(ii) Find the distance between q(x) = 2x
2
2
please turn over
SECTION B continued
B3. (i) (a) Explain what is meant by two square matrices A, B being similar. [1]
(b) Prove that similar matrices have the same characteristic polynomial. [4]
(ii) Let
A =
(a) Determine the eigenvalues of A. [4]
(b) Determine the eigenspaces of A. [7]
(c) Write down a basis for R
3 consisting of eigenvectors of A. [2]
(d) Write down an invertible matrix P and a diagonal matrix D such that P
− 1 AP = D. [2]
B4. Let V be a Euclidean space with inner product 〈 , 〉. Prove that
(i) ||x + y|| ≤ ||x|| + ||y|| for all x, y ∈ V (you may use without proof that |〈x, y〉| ≤
||x||.||y||); [5]
(ii) ||x + y||
2
2 = 2(||x||
2
2 ) for all x, y ∈ V ; [3]
(iii) 〈x, y〉 = 0 implies that ||x||
2
2 = ||x + y||
2 for all x, y ∈ V ; [1]
(iv) if e 1 ,... , en is an orthonormal basis for V and x ∈ V then
x = α 1 e 1 +... αnen where αi = 〈x, ei〉, and
||x||
2 = αα
T where α = (α 1 ,... , αn); [7]
(v) if T : V → V is a Euclidean transformation, then 〈T (x), T (y)〉 = 〈x, y〉 for all x, y ∈ V. [4]
end of exam
