M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 1
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 R such that (, 6) lies on L
(iv) the point of intersectio
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 4
The question for discussion with the personal tutor is Question 1
1*. 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.
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 5
The question for discussion
1 2 3
0 1 2
1. (a) Calculate det
3 0 1
2 3 0
with the personal tutor is Question 6
0
3
.
2
1
t1
3
3
(b) Solve for t the equation det 3 t + 5 3 = 0.
6
6
t4
a
bx cx
(c) Sol
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 3
The question for the discussion with the personal tutor is Question 5
2
3
1. Let A =
1
(i) Solve the
1 3
x1
and x = x2 .
1 4
x3
1 2
system Ax = 0.
1
(ii) Let b = 2 . Find all solutions of the sy
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 6
The question for discussion with the personal tutor is Question 2.
1. Decide which type of conic is given by each of the following equations
by reducing the equation to standard form.
(i) x1 x2 = 2
(
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 7
1. (i) Find the (vector) equation of the line joining the points (3, 1, 2)
and (6, 1, 8). Find the equation of the plane containing this line and the
point (0, 2, 1).
(ii) Find the equation of the pl
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 8
Notation: when A, B are square matrices of the same size we write [A, B]
for AB BA. The trace of a square matrix A is dened as the sum of
diagonal entries tr(A) = a11 + . . . + ann . An n n matrix A
re/0V r AprTVFHPman
r) mono? mSLF\rFFrPF. 0,1. no a, 09FPor 3 Cir fir o!
ELY??? +0.5)». Cbxm. rpbrmv it? ght? in? S .
E or x. .r. FA: EFL ?%§r.&1M9+vsnvm.¢ 7P9?
T. urrwuk fcrmxc CA. 3.9%? 3 EC g. 91.»; Pa
Draw» rrlh F+Oh<lr>vmr F
\
cr+ 477;»;
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 9
The question for discussion with the tutor is Question 6.
1. Quaternions are 4-dimensional vectors (x1 , x2 , x3 , x4 ) R4 , with the
usual addition and multiplication by scalars. We write such a vec