2B  Linear Algebra (2012)
Exercise Sheet 5
1. In this question we consider the following maps:
h : R3 R2 given by h(x, y, z) = (7x, y z).
k : R3 R2 given by k(x, y, z) = (2x + 3y z, 3x + 2y + 2z).
In each case:
(1) write down the matrix A dening the ma
Axioms (summary)

A field of scalars:
Set of elements (scalars) F
Closed under addition and multiplication
The above are needed to complete the following conditions:
Associativity brackets can be modified freely when adding or multiplying
Zero and unit
Functions (Summary)

A function is denoted for example as f: X>Y
X is called the domain of f
Y is called the codomain of f
Its a map or a mapping transformation
Given f: X>Y and h:X>Z we can form a composition
The rule of composition is g f(x) = g(f(x
2B  Linear Algebra (2012)
Exercise Sheet 4
A shorter exercise sheet this time (since the class test takes place on Friday 26th October)
covering only the material from Lectures 12, 13, 14.
Let F be a eld and let U, V, W be vector spaces over F .
1. Let T
2B  Linear Algebra (2012)
Exercise Sheet 2
The questions on this exercise sheet cover the content from lectures 4, 5, 6, 7.
1. Which of the vectors (0, 3, 6), (3, 9, 2), (0, 0, 0) and (7, 8, 9) can be written as a linear
combinations of u = (2, 1, 4), v
2B  Linear Algebra (2012)
Exercise Sheet 3
1. Provide a much simpler solution to Q9 from Exercise Sheet 2 in light of what you learned in
Lecture 8. That is, decide which of the following sequences is a basis for R4 :
S1 : (1, 0, 1, 1), (2, 0, 0, 3);
S2
2B  Linear Algebra (2012)
Exercise Sheet 6
1. Given that the polynomial p(t) = t3 8t2 + 22t 20 has 2 as one of its roots, nd all the
complex roots of p(t). [We introduce this question so that you remind yourself what you used
to know about nding roots of
2B  Linear Algebra (2012)
Conclusions from the Class Test
Overall conclusions. The class test included 4 marks for giving denitions, and 10 marks for
reproducing two of the six proofs that we listed as being examinable. This means that nearly
50% of the
2B  Linear Algebra (2012)
Instructions for the Degree Exam
The degree exam will test your understanding of the content from all 22 of the lectures, and
exercise sheets 1 to 6. However, the situation isnt as bad as it sounds because:
Denitions. 20% of the
2B  Linear Algebra (2012)
Instructions + Sample questions for the Class Test
Instructions for the class test. The class test for 2B will take place during your normal
lecture hour on Friday 26th October. Note however that not everyone can take the class
2B  Linear Algebra (2012)
Exercise Sheet 1
1. Consider the matrices
2 6 8
A = 0 5 3 ;
4 7 9
2 6 8
B = 0 5
3 ;
0 5 25
C=
1 0
;
5 2
D=
2 4
.
1 2
(i) Find elementary matrices E1 and E2 such that E1 A = B and E2 B = A.
(ii) Write the matrix C as a product of