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
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M1M2
imperial College
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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.
Mathematical Methods ll
Date: Wednesday, 23 May 2012. Time
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 (MATHEMATICS)
May 2012
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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, 1, 0) + a(1, 1, 1).
(c) The system reduces to echelon
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 v3 = 0, do not span R3
(iii) linearly independent, span
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 expressions
over i = 1, . . . , n is symmetric under swappin
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 vector b a = (2, 1),
hence normal (1, 2). Hence its equation
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
and
(any non-zero real numbers a, b). So e.g. P =
2b
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 0B 1 = A1 (AB)B 1 = (A1 A)(BB 1 ) = I,
which is visibl
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, 1, 1)| R.
2. Write p a = h + u, where h is perpendicul
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
the equation becomes y12 y22 = 4, or y22 /4 y12 /4 = 1, a
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, b R.
(d) General solution (5a, 2a, a) for any a R.
2.
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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 College of Science.
Mechanics
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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 Methods H
Date: examdate Time: examtime
Credit will be giv
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
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 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 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 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 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 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
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.
Mathematical Methods |
Date: Friday. 24 May 2013. Time:
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.
Mechanics
Date: Thursday. 23 May 2012. Time: 10.00am. Time
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.
Mathematical Methods H
Date: Thursday. 19 May 2011. Ti