1. Eigenvalues and Eigenvectors
To facilitate the study of eigenvalues and eigenvectors in depth, we need to
consider complex matrices and vectors. Let
z1
.
Cn = . z1 , . . . , zn C
.
zn
be the complex vector space of n 1 complex matrices, in which ve
1. Linear transformations
Linear transformations are the bread and butter of Linear Algebra. You have
already encountered them in Geometry I. Roughly speaking a linear transformation
is a mapping between two vector spaces that preserves the linear structu
1
0.1
Determinants
Let A = (aij ) be a 22 matrix. Recall that the determinant of A was dened
by
a
a
det(A) = 11 12 = a11 a22 a21 a12 .
(1)
a21 a22
Notation 0.1.1. For any n n matrix A, let Aij denote the submatrix
formed by deleting the i-th row and the j
1
0.1
(Linear) span of vectors
Denition 0.1.1. Let v1 , . . . , vn be vectors in a vector space V . A linear
combination of v1 , . . . , vn is a vector v of the form
v = 1 v1 + + n vn
where 1 , . . . , n are scalars. The set of all linear combinations of
1
0.1
Revision from Geometry I
Recall that an mn matrix A is a rectangular array of scalars (real numbers)
a11 a1n
.
. .
.
.
.
.
am1 amn
We write A = (aij )mn or simply A = (aij ) to denote an m n matrix whose
(i, j)-entry is aij , i.e. aij is the i-th
1
Basic terminology and examples
A linear equation in n unknowns is an equation of the form
a1 x 1 + a2 x 2 + + an x n = b ,
where a1 , . . . , an and b are given real numbers and x1 , . . . , xn are variables.
A system of m linear equations in n unknowns
MTH5112 Linear Algebra I 2011
Coursework 9 Solutions
Exercise 1. The Gram Schmidt process starts by setting
1
0
v1 = x1 = .
1
0
The vector v2 is constructed by subtracting the orthogonal projection of x2 onto Span (v1 ) from
x2 , that is,
1
0
4
x2 , v
School of Mathematical Sciences
Mile End. London E1 4N5
MTH5112 Linear Algebra l
MID-TERM TEST
Date: 12 November 2010 Time: 11.0041.40
Surnames A to C
Surnames D to Z
Complete the following information:
The duration of the test is 40 minutes. Answ
School of Mathematical Sciences
Mile End, London E1 4N5
MTH5112 Linear Algebra I
TEST
Date: 10 November 2011 Time: 3.00w3.40pm
3
Arts 2 Lecture Theatre 3 Surnames A to M
l V NJ
| m
I Mason Lecture Theatre Surnames N to Z
i
Complete the following inf
School of Mathematical Sciences
Mile End, London E1 4NS
MTH5112 Linear Algebra I
MID-TERM TEST
Date: 12 November 2010 Time: 11.0011.40
FB328
Surnames A to C
Octagon
Surnames D to Z
Complete the following information:
Name
Student Number
(9 digit code)
The
MTH5112 Linear Algebra I 2011
Coursework 9
(Late Coursework or Coursework put in the wrong box will NOT be marked.)
Hand in your solution of the starred exercises by 4.30pm, Thursday 8 December 2011.
Put it in the Red Linear Algebra I Collection Box in th
MTH5112 Linear Algebra I 2011
Coursework 8 Solutions
Exercise 1.
(a) L is not linear since L(0) = (1, 2, 3)T = 0.
(b) L is linear, since, if x, y R3 and R, then
(i) L(x + y) =
(ii) L(x) =
(x3 + y3 ) (x2 + y2 )
(x2 + y2 ) (x1 + y1 )
x3 x2
x2 x1
=
=
x3 x2
x
MTH5112 Linear Algebra I 2011
Coursework 8
Hand in your solution of the starred exercises by 4.30pm, Thursday 1 December 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put in
MTH5112 Linear Algebra I 2011
Coursework 6 Solutions
Exercise 1.
(a) We prove by contradiciton. Suppose v, v1 , . . . , vn are linearly dependent. Then there are
scalars , 1 , . . . , n , not all 0, such that
v + 1 v1 + + n vn = 0 .
Since v1 , . . . , vn
MTH5112 Linear Algebra I 2011
Coursework 7
Hand in your solution of the starred exercises by 4.30pm, Thursday 24 November 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put i
MTH5112 Linear Algebra I 2011
Coursework 5 Solutions
Exercise* 1. An n n matrix A is called invertible if there is a matrix B Rnn such that
AB = BA = I, where I is the identity matrix.
(a) Let AB = I, as given. Then 1 = det I = det(AB) = (det A)(det B) im
MTH5112 Linear Algebra I 2011
Coursework 6
Hand in your solution of the starred exercises by 4.30pm, November 17 November 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put i
MTH5112 Linear Algebra I 2011
Coursework 4 Solutions
Exercise 1. A is invertible i det A = 0.
Since the determinant of a matrix equals the determinant of its transpose, we have A I is
invertible i det(A I) = 0 i det(A I)T ) = 0 i det(AT I) = 0 i AT I is i
MTH5112 Linear Algebra I 2011
Coursework 5
Hand in your solution of the starred exercises by 4.30pm, Thursday 3 November 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put in
MTH5112 Linear Algebra I 2011
Coursework 4
Hand in your solution of the starred exercises by 4.30pm, Thursday 27 October 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put in
MTH5112 Linear Algebra I 2011
Coursework 3 Solutions
Exercise 1. Let A =
a b
c d
and, let P and Q be as given in the question. Then
AP =
a b
c d
1 0
0 0
=
a 0
c 0
= PA =
a b
0 0
which implies b = c = 0. Now
AQ =
0 a
d 0
= QA =
0 d
a 0
implies a = d and he
MTH5112 Linear Algebra I 2011
Coursework 2 Solutions
Exercise 1.
A=
1 0
, B=
0 0
0 0
, P =
0 1
0 0
, S = 2P I
0 1
Exercise 2. Let A = (aij ) and B = (bij ). We dene AT = (aji ).
(a) Since (A + B)T and AT + B T have the same size we only need to check that
MTH5112 Linear Algebra I 2011
Coursework 3
Hand in your solution of the starred exercises by 4.30pm, Thursday 20 October 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
(Late Coursework or Coursework put in
MTH5112 Linear Algebra I 2011
Coursework 2
Hand in your solution of the starred exercises by 4.30pm, Thursday 13 October 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
Write your name and student number on
MTH5112 Linear Algebra I 2011
Coursework 1 Solutions
Exercise 1. In row echelon form: a, c, e, f; in reduced row echelon form: c, f.
Exercise 2.
(a) A possible example is
1 1 2 3
0 1 1 0 .
0 0 0 0
(b)
(i) The augmented matrix of the system is
1 0
4
1 6
0
B. Sc. Examination by course unit 2011
MTH5112
Linear Algebra I
Duration: 2 hours
Date and time: 26 May 2011,
2.30pm
Apart from this page, you are not permitted to read the contents of
this question paper until instructed to do so by an invigilator.
You s
MTH5112 Linear Algebra I 2011
Coursework 1
Hand in your solution of the starred exercise by 4:30pm, Thursday 6 October 2011.
Put it in the Red Linear Algebra I Collection Box in the (Mathematical Sciences) basement.
Write your name and student number on y
MTH5112 Linear Algebra I 20122013
Coursework 11 Solutions
Exercise 1.
(a) First we show that N (A) N (AT A). In order to see this suppose that x N (A), that is,
Ax = 0.Then AT Ax = AT 0 = 0, that is, x N (AT A).
Next we show that N (AT A) N (A). In order
MTH5112 Linear Algebra I 20122013
Coursework 9 Solutions
Exercise 1.
(a) Let E be the standard basis for R3 . Since
3
3
L(e1 ) = 4 , L(e2 ) = 5 ,
3
6
3
L(e3 ) = 6 ,
2
the matrix representation of L with respect to the standard basis is
3
3
3
[L]E = 4