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 addi
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imperial College
London
BBC. MSci and MSc EXAMINATIONS (MATHEMATICS)
May June 2012
This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science.
Mat
IMPERIAL COLLEGE LONDON
Course:
M1A1
Setter:
Professor Andrew Parry
Checker: Dr Daniel Moore
Editor:
Dr Dmitry Turaev
External: Professor Paul Houston
Date:
April 3, 2012
BSc and MSci EXAMINATIONS (MA
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 3
1. (a) The general solution is x = a(1, 1, 1) for any a R.
(b) We spot that (1, 1, 0) is a solution to Ax = b. Thus the general
solution is (1,
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 9
1. All the claims are direct consequences of defintions.
2. (a) (i) linearly independent, do not span R3
(ii) linearly dependent since v1 + v2
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 8
1. Both identities follow straight from the definition of the bracket.
2. The ii-entry of AB is a1i bi1 +. . .+ani bin . The sum of these expre
M1GLA Geometry and Linear Algebra 2014
Solutions for Sheet 1
1. (i) L = cfw_(1, 2) + (3, 3)| R
(ii) n = (1, 1)
(iii) = 3
(iv) ( 34 , 53 )
2. (i) The line is cfw_a + (b a)| R, which has direction vecto
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 5
1. (a) 96; (b) solutions t = 2, 4; (c) x = 0, b c, (a + b + c)/2.
a
2. (i) Eigenvalues are 1 and 3. The corresponding eigenvectors are
a
b
1
1
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 4
1. (i) False for any pair A, B such that AB 6= BA.
0 1
(ii) False, e.g. for A = B =
.
0 0
(iii) True. Assume A1 and B 1 both exist, then
0 = A1
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 7
1. (i) The line is cfw_(3, 1, 2)+(3, 0, 6)| R. The plane is 2x1 +5x2 x3 =
9.
(ii) The plane is 2x1 x2 + x3 = 1. The line is cfw_(1, 2, 1) +
(2,
M1GLA Geometry and Linear Algebra 2014
Solutions Sheet 6
1. (i) Rotate axes (anticlockwise) through/4. If (y1 , y2 ) are
the new
coordinates, then we have x1 = (y1 y2 )/ 2, x2 = (y1 + y2 )/ 2, and
th
M1GLA Geometry and Linear Algebra 2014
Solutions for Sheet 2
1. (a) Unique solution (2, 1, 3, 1).
(b) No solution
(c) General solution (x1 , . . . , x5 ) = (4 a, 10 25 b 3a, 2 + 12 b, b, a) for
any a,
MlAl
CPRRECTEQ VEES;O~I
Imperial College
LQMQSE
BSc. MSci and MSC EXAMINATIONS (MATHEMATlCS)
May H June 2014
This paper is also taken for the relevant examination for the Associateship of the
Royal Co
Imperial College
BSc and MSci EXAMINATIONS (MATHEMATICS)
MayJune 2014
This paper is also taken for the relevant examination for the Associateship of the Royal College of
Science.
M1M2
Mathematical Met
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
diagon
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 an
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
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
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
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 s
M1GLA Geometry and Linear Algebra 2014
Exercise Sheet 2
1. Find all solutions of the following systems of linear equations:
(a)
x1 2x2 + x3 x4
3x1 6x2 + 2x3
x3 2x4
2x1 3x2 + 3x4
(c)
x1 2x3 + x4 + x5 =
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
M1M2
Imperial College
London
BSc. MSci and NBC EXAMINATIONS (MATHEMATICS)
May June 2013
This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science.
Mat
MlAl
Imperial College
London
BSc, MSci and MSC EXAMINATIONS (MATHEMATICS)
May June 2012
This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science.
Mec
M1M2 i
Imperial College
London '
BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)
May June 2011
This paper is also taken for the relevant examination for the Associateship of the
Royal College of Science.