M1GLA Geometry and Linear Algebra
Introductory Problem Sheet
(not for assessment)
1. In the world famous East Grinstead Zoo, the elephants and tigers all live in
the same enclosure. Their relationships obey the following rules:
(1) There is at least one e
M1GLA Geometry and Linear Algebra
Solutions to Problem Sheet 9
1. Not a (signicant) vector space, eg for v = Liebeck there is no negative vector
v such that v (v) = Liebeck.
2. (i) 0 = (0 + 0) = 0 + 0 by axiom S1. Subtracting 0 from both sides
gives 0 = 0
M1GLA Geometry and Linear Algebra, Solutions to Sheet 7
10
3
1. (i) This is xT Ax = 4, where A =
3
. Evalues of A are 11,1, unit evectors
2
3
1
, v2 = 1
. So if P = (v1 v2 ) then the change of variables x = P y
10
1
3
2
2
reduces eqn to 11y1 + y2 = 4, an
M1GLA Geometry and Linear Algebra, Solutions to Sheet 8
1. (a) Line is cfw_(3, 1, 2) + (3, 0, 6) : R. Plane is 2x1 + 5x2 x3 = 9.
(b) Plane 2x1 x2 + x3 = 1. Line cfw_(1, 2, 1) + (2, 1, 1) : R.
2. Same proof as for R2 , see Sheet 2, Q7. Distance
6.
3. (i) S
M1GLA Geom and Linear Algebra, Solutions to Sheet 6
1. (i) Evalues 1,3. Corresponding evectors
a
a
and
b
2b
(any non-zero
1
1
real numbers a, b). So eg. P =
will do.
1 2
a
b
c
1 1 1
(ii) Evals 1,2,3. Evecs 0 , b , c (any a, b, c = 0). P = 0 1 1
0
0
c
0
M1GLA Geometry and Linear Algebra, Solutions to Sheet 5
1. (i) False for any pair A, B such that AB = BA, eg A =
0
1
1
0
0
1
,B =
1
.
0
(ii) False, eg A = B =
0
0
1
.
0
(iii) True: assume A, B both invertible and AB = 0. Then 0 = A1 (AB) =
(A1 A)B = B, co
M1GLA Geometry and Linear Algebra, Solutions to Sheet 4
1. Let p be a particular solution, so Ap = b. If v is any solution, then Av = b, so
A(v p) = Av Ap = b b = 0, so v p is a solution of Ax = 0. If we write h = v p
then v = p + h. So every soln to Ax =
M1GLA Geometry and Linear Algebra, Solutions to Sheet 3
1. Suppose x.n = a.n. Then (xa).n = 0. Writing y = xa this says (y1 , y2 ).(u2 , u1 ) =
0, so y2 u1 y1 u2 = 0. In lectures (proof of Cauchy-Schwarz) we showed that this implies
that y is a scalar mul
M1GLA Geometry and Linear Algebra Exercise Sheet 9
1. At a meeting of the Imperial Mathematics Department sta, Prof Mestel announces that
he has discovered a signicant new vector space. The vectors are the members of sta, with
scalar multiplication v = v
M1GLA Geometry and Linear Algebra
Exercise Sheet 8
1. (a) Find the (vector) equation of the line joining the points (3, 1, 2) and (6, 1, 8). Find the
equation of plane containing this line and the point (0, 2, 1).
(b) Find the equation of the plane normal
M1GLA Geometry and Linear Algebra
Exercise Sheet 7
1. Reduce the following conics to standard form by rotation and translation, and sketch each
one in the x1 x2 -plane:
(i) 10x2 + 6x1 x2 + 2x2 4 = 0
1
2
(ii) x2 + 8x1 x2 + 7x2 + 18 5x1 + 36 = 0
1
2
(iii) x
M1GLA Geometry and Linear Algebra
Exercise Sheet 6
1*. (For d with personal t) For each of the following matrices A, nd the eigenvalues and
eigenvectors of A, and then either nd an invertible matrix P such that P 1 AP is diagonal, or
prove that no such ma
M1GLA Geometry and Linear Algebra
Exercise Sheet 5
1*. (For d with personal t) Let A and B be n n matrices with real entries. For each of the
following statements, either give a proof or nd a counterexample. (To nd counterexamples,
consider n = 2.)
(i) A2
M1GLA Geometry and Linear Algebra
Exercise Sheet 4
1. Prove that all solutions of a system Ax = b are of the form p + h, where p is a particular
solution and h is a solution of Ax = 0.
2 1 3
c1
1
x1
2. Let A = 3 1 4 , let x = x2 and let b = 2 and c = c2 .
M1GLA Geometry and Linear Algebra
Exercise Sheet 3
Starred question for discussion with Personal T
1. Let L be the line cfw_a + u : R, and let n = (u2 , u1 ) be a normal to L. Show that
every point x R2 satisfying the equation x.n = a.n lies on the line L
M1GLA Geometry and Linear Algebra
Exercise Sheet 2
1. Let a, b be the points (1, 2), (2, 5). Find
(i) in vector form, the line L through a and b
(ii) a vector perpendicular to L
(iii) a scalar such that (, 6) lies on L
(iv) the point of intersection of L
M1GLA Geometry and Linear Algebra, Solutions to Sheet 1
1. (a) There is an elephant, by (1). He likes at least 2 tigers by (3), and there is a further
tiger by (4). Hence there are at least 3 tigers.
(b) Say the 3 tigers are T1 , T2 , T3 . For each pair i
M1GLA Geometry and Linear Algebra
Bumper Christmas Problem Sheet 10
1. (a) In each of the following cases, say (giving reasoning) whether or not the given vectors are
linearly independent, and whether or not they span the whole of R3 :
(i) (5, 3, 0), (2,