Department of Mathematics
MATHS 208 Assignment 2 Due: 4pm Tuesday 3 May 2016
• For each question you must show your working whenever it is required in order to get full marks.
• When working is required answers only will not be awarded full marks.
• There is one exception. If you need to row reduce a matrix you are only required to give the reduced matrix with no working. You may use any application you like to obtain it.
1. (6 marks)
(a) Suppose that
v1 =
10
1
, v2 =
−4−6 −10
, and v3 =
02
2
.
Either show that the collection of vectors {v1, v2, v3} is linearly dependent, or show that it is linearly independent.
(b) Suppose that
u1 =
0 −3 −3 0
, u2 =
0 3 0 3
, u3 =
1 0 1 2
and u4 =
−2 0 −1 −1
.
Either show that the collection of vectors {u1, u2, u3, u4} is linearly dependent, or show that it is linearly independent.
2. (6 marks)
Let
v1 =
1 4 2 −2
, v2 =
1 2 4 0
, v3 =
−2 −6 −6 2
, v4 =
−1 −2 −4 0
, and v5 =
1 1 −1 −1
,
and let
A = [v1 v2 v3 v4 v5] =
1 1 −2 −1 1 4 2 −6 −2 1 2 4 −6 −4 −1 −2 0 2 0 −1
∼
1 0 −1 0 0 0 1 −1 −1 0 0 0 0 0 1 0 0 0 0 0
.
(a) Find a basis for col(A), the column space of matrix A.
(b) Express the vector v3 as a linear combination of the basis vectors found in (a).
3. (7 marks)
Suppose
A =
1 3 0 −1 20 −2 4 −2 0
2 5 3 −4 0
∼
1 0 6 −4 20 1 −2 1 0
0 0 1 −1 −4
.
(a) Write down a basis for col(A). What is the rank of A?
MATHS 208 Assignment 2 Page 1 of 3
(b) What is the set of all possible vectors b such that the matrix equation Ax = b has a solution.
4. (12 marks)
(a) Let
A =
1 −4 00 −6 2
1 −10 2
∼
1 0 −430 1 −1
3 0 0 0
.
(i) Find a basis for null(A).
(ii) Consider the linear system 1 −4 00 −6 2
1 −10 2
x1x2 x3
=
−12
1
.
Find one solution to this system.
(iii) Find the general solution to the linear system in (ii), expressed as a vector plus a vector space.
(b) Consider the linear system
1 2 −2 −1 0 2 3 −5 1 −7 0 1 1 −1 1 −1 1 5 −1 0
x1 x2 x3 x4 x5
=
3 4 2 3
.
Given that
1 2 −2 −1 0 ∣∣ 3
2 3 −5 1 −7 ∣∣ 4
0 1 1 −1 1 ∣∣ 2
−1 1 5 −1 0 ∣∣ 3
∼
1 0 −4 0 1 ∣∣ −1
0 1 1 0 −2 ∣∣ 2
0 0 0 1 −3 ∣∣ 0
0 0 0 0 0 ∣∣ 0
,
find the general solution to this linear system. Express your answer in the form of a vector plus a vector space.
5. (4 marks)
Let
A =
1 2 3 4 5 60 0 0 1 2 3
0 0 0 0 1 1
.
(a) State the dimension of the column space of A.
(b) State rank(A).
(c) State the nullity of A.
(d) Which of the sets: col(A), null(A) and the general solution to Ax = b, are vector spaces?
6. (12 marks)
(a) Is the set of vectors: {( 1 2
) ,
( 3 4
) ,
( 5 6
)} orthogonal? Give a reason for your answer.
MATHS 208 Assignment 2 Page 2 of 3
(b) Suppose S is the vector space spanned by the set of vectors {u1, u2}, where
u1 =
−1 −1 −1 1
and u2 =
1 2 1 1
.
(i) Use the Gram-Schmidt process to obtain an orthonormal set of vectors {v1, v2} that spans S.
(ii) Can the vector
1 1 1 1
be expressed as a linear combination of v1 and v2?
7. (2 marks)
Challenge Question: Consider the set V of all polynomials with real coefficients – this set is a vector space over R if operations of addition and scalar multiplication are defined in the “standard” way. (You don’t have to prove that V is a vector space – take it as a given). Consider a subset W of V consisting of all polynomials of degree less or equal to 2. Is W a subspace of V ? Can you give an example of a basis for W ? What is the dimension of W ? State clear reasons for all your answers.
8. (1 marks)
Write a paragraph stating which question from this assignment you found most challenging and why.
MATHS 208 Assignment 2 Page 3 of 3
