SOM201 2008 Solution 1
(i) The rank of A (which is equal to the column rank of E) is 4.
(ii) The general solution is
x7 = 0, x6 = , x5 = , x4 = + , x3 = , x2 = 2 2, x1 = + ,
where , , are arbitrary real
AX = 0 is
numbers. Thus, in (column) vector fo
MAS201 2009 Solution 1
(i) We have
1
0
A
0
0
1
0
1
2
So
1
0
3
6
1
0
2
4
1
0
A E =
0
0
1
0
0
0
1
2
0
0
0
1
0
0
1
1
0
0
1
2
0
0
1
3
0
0
2
3
0
0
(ii) The rank of A (which is equal to the column rank of E) is 2.
(iii) Since the rank of A is less
MAS201 PROBLEM SHEET 3
Lecture 5
Exercise 1. You should justify your answers to the following questions.
3
2
4
(a) Is the list a1 =
,a =
,a =
a basis for R2 ?
5 2 3
7
4
9
8
3
(b) Is the list b1 = 8 , b2 = 7 , b3 = 2 a basis for R3 ?
1
6
7
5
7
1
1
3
8
(c)
Linear Mathematics for Applications
Exam
(1) You may assume the following row reductions. Some of them are relevant, and some
are not. Some questions can be done more easily without row-reduction.
1 0
0
0 1 1
1 0 0 0 9
0
2 0 1 10
0 1
2 1 3 3
0 1 0 0 8
Linear Mathematics for Applications
Exam
(1)
(a) Which of the following
answers. (3 marks)
1 2
A = 0 0
0 0
matrices are in reduced row echelon form (RREF)? Explain your
0
0
1
3
0
4
0
B = 0
1
0
0
0
0
1
0
(b) Row-reduce the following matrix. (6 marks)
11 1
MAS201 PROBLEM SHEET 5
Lecture 9
4 1
Exercise 1. Consider the matrix A =
. Find an invertible matrix U and a diagonal matrix D
6 9
such that A = U DU 1 . Check directly that the equation A = U DU 1 holds.
Solution: The characteristic polynomial is
A (t) =
MAS201 PROBLEM SHEET 7
Lecture 13
Exercise 1. Consider the following web of pages and links.
3
4
2
5
9
1
6
8
7
Let a be the PageRank of page 1, and let b be the PageRank of page 9. By symmetry, pages 2 to 8 must
also have rank a. Use the consistency and n
MAS201 PROBLEM SHEET 10
Exercise 1. Show that if A is an orthogonal matrix then det(A) = 1.
Solution: By the denition of an orthogonal matrix we have AT A = I, so det(AT ) det(A) = det(I) = 1.
For any square matrix A we have det(AT ) = det(A), so we now s
MAS201 PROBLEM SHEET 8
Lecture 15
Exercise 1. Let V be the set of vectors of the form
v = 2p q
q+r
3p
r
T
(where p, q, and r are arbitrary real numbers). Find a list of vectors whose span is V .
Solution: This is similar to examples 19.16 and 19.17. The g
MAS201 PROBLEM SHEET 6
Lecture 11
Exercise 1. Solve the following system of dierential equations using the method in Section 15:
x = 2x 3y
x = 1 when t = 0
y = 3x 2y
y = 0 when t = 0.
This involves complex eigenvalues. You should remember the rules
cos(t)
MAS201 PROBLEM SHEET 4
Lecture 7
Exercise 1. Calculate the determinant of the matrix
a 0
d 0
A=
e f
i 0
b
0
g
0
c
0
h
j
Solution: The most obvious approach is to expand along the top row. This gives
det(A) = a det(B1 ) 0 det(B2 ) + b det(B3 ) + c det(B4 )
MAS201 PROBLEM SHEET 2
Lecture 3
Exercise 1. Put
p1 =
1
2
p2 =
3
6
p3 =
2
.
4
Describe geometrically which vectors in R2 can be expressed as a linear combination of p1 , p2 and p3 .
Give an example of a vector that cannot be described as such a linear com