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

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