School of Technology
Department of Mathematical Sciences
8620 Linear Algebra and Analysis: Coursework II
1.
2.
Consider the sequence
an  defined by an 1 
an
9

2 2 an
i 
Show that
 ii 
Show that an  is a decreasing sequence.
 iii 
Hence, find
a1  4 .
an  3 n  .
lim an .
[10 marks]
n 
Suppose that f is a function defined on an interval
f  x   f t   x  t
x, t  a, b .
2
Prove that f is constant on
a, b and that
 a, b .
[10 marks]
3.
  3  2 x 2  sin  4 x   12 x 
.
Use L’Hôpital’s rule to evaluate lim 
x 0  2  3 x 2 sin 5 x  10 x 






4.
Use the Comparison Tests for Integrals (Haggarty, page 248 and Question 5 page 251)
to show that
MKP, 28 July 2017


0
1
1


x 1  x 5 


1
4
4
dx exists.
[10 marks]
[10 marks]
5.
Find a basis for the kernel, and a basis for the range, of the linear transformation,
5
T:

4
, given by
T ( x1 , x2 , x3 , x4 , x5 )
 ( x1  2 x2  x4  x5 , 2 x1  x2  3x3  x4 ,  x1  2 x3  x5 , 2 x4  8x5 ) .
Hence verify in this case that the Rank-Nullity Theorem holds.
6.
[10 marks]
  {(1,3), (2,5)} and   {(1,1), (4,1)} are bases for 2 , find the change-of-basis
2
 2 is the linear
matrices E  I  ,   I E and   I  . Hence find  T  , where T :
If
2
2
transformation given by
the vector
v  (1,1) 
T ( x1 , x2 )  (6 x1  x2 , 4 x1  3x2 ) . Verify that
2

T  v   Tv 
for
.
[10 marks]
7.
i 
Show that the matrix,
 1  1 1 


A   1 3 1 ,
  3 1 1 


is diagonalizable, and find a diagonal matrix D and invertible matrix P such that
 ii 
Let T :
3

3
P1 AP  D .
be the linear transformation given by
T ( x1 , x2 , x3 )  ( x1  x2  x3 , x1  3x2  x3 , 3x1  x2  x3 ) .
Using your result for part  i  , state a basis B for
3
such that the matrix
B
T B is diagonal.
[10 marks]
8.
i 
Show that the matrix,
 4 0 0


B  1 4 0 ,
 0 0 5


is not diagonalizable.
 ii 
 iii 
Show that if a matrix A is diagonalizable and invertible, then so is
Let
A1 .
C be a diagonalizable matrix, and suppose that D  P 1CP , where P is invertible
1
and D is diagonal. Show that D  P C P , for every positive integer
k
k
k.
[10 marks]
MJG, 28 July 2017