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LANCASTER UNIVERSITY
2008 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. Write down, in Cartesian form, the equations of the straight line through the point with
position vector (3, 1 , −2) parallel to the direction given by (1, 0 , 1). [5]
A2. (i) Find the reduced echelon form E for the matrix
A =
[5]
(ii) Find an invertible matrix P such that E = P A. Check your answer. [5]
A3. Let S = {(1, − 3 , 1), (− 2 , 6 , −2), (2, 1 , −4), (− 1 , 10 , −7)}.
Find the dimension of sp S, the subspace spanned by S. [10]
A4. Let S = {(x, y, 2 y − x) : x, y ∈ R}. Show that S is a subspace of R
3
. [5]
A5. Find the matrix of the linear transformation T : R
2 → R
2 given by
T ((x, y)) = (2x + y, 3 x − y)
with respect to
(i) the standard basis in both domain and codomain; [1]
(ii) the basis (1, 1), (2, −3) in both domain and codomain. [4]
please turn over
SECTION A continued
A6. Show that (1, 1 , −2) and (1, 1 , −1) are eigenvectors for the matrix
A =
and find the corresponding eigenvalues. [5]
A7. For each v = (x 1 , y 1 ), w = (x 2 , y 2 ) ∈ R
2 define
〈v, w〉 = 3x 1 x 2 + 2y 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
B3. (a) Let^ A^ = [aij ] be an^ m^ ×^ n^ matrix with elements in a field^ F^.
(i) Give a criterion involving rank A and rank (A|B) for the system of linear equations
AX = B to be consistent. [1]
(ii) Let X = X 1 be a fixed solution to the system AX = B. Show that if X = X 2 is a
solution to the system AX = 0 then X = X 1 + X 2 is a solution to AX = B. Show
furthermore that every solution to AX = B is of this form. [6]
(b) (i) Find the reduced echelon form for the matrix C where
C =
and hence determine its rank. [8]
(ii) Use (a)(i) above to show that the system
2 z +w = 9
− 2 x − 6 y + 2z = 2
2 x + 6y − 2 z +2w = 0
3 x + 9y + 2z +2w = 19
is consistent, and give its general solution. [5]
B4. Let
A =
(i) Determine the eigenvalues of A. [6]
(ii) Determine the eigenspaces of A. [9]
(iii) Find an orthonormal basis for R
3 consisting of eigenvectors of A. [3]
(iv) Write down an orthogonal matrix P and a diagonal matrix D such that P
T AP = D. [2]
end of exam
