SIT292 LINEAR ALGEBRA 2017


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 SIT292 LINEAR ALGEBRA 2017

Assignment 3

Due: 5 p.m. September 28, 2017

Note that by University regulations, the assignment must reach the unit chair

by the due date, even if it is posted.

  1. State the de_nition of the row-rank. For the following matrix

A =

2

4

1 2 􀀀3 0

2 4 􀀀2 2

3 6 􀀀4 3

3

5

(a) determine the row-rank.

(b) _nd a set of generators for the row space of A.

(c) _nd a basis for the row space of A. Explain why it is a basis.

( 4 + 2 + 4 = 10 marks)

  1. For the following matrix

2

664

0 2 0

1 0 1

0 2 0

3

775

(a) _nd the eigenvalues

(b) _nd the eigenvectors corresponding to these eigenvalues

(c) starting with the eigenvectors you found in (a) construct a set of

orthonormal vectors (use the Gram-Schmidt procedure).

( 5 + 10 + 5 = 20 marks)

  1. The set of ordered triples f(1; 0; 1); (􀀀1; 1; 1); (0; 1; 0)g forms a basis

for R3. Starting with this basis use the Gram-Schmidt procedure to

construct an orthonormal basis for R3.

( 10 marks)

  1. Denote by Rn the set of all n-tuples of real numbers. Rn is called

the Euclidean vector space, with equality, addition and multiplication

de_ned in the obvious way. Let V be the set of all vectors in R4

orthogonal to the vector (0; 1;􀀀2; 1); i.e. all vectors v 2 V so that

vT (0; 1;􀀀2; 1) = 0.

(a) Prove that V is a subspace of R4.

(b) What is the dimension of V (provide an argument for this), and

_nd a basis of V . (Hint: observe that the vector (0; 1;􀀀2; 1)

does not belong to V , hence dim V _ 3; next _nd 3 linearly

independent vectors in V .)

(10 + 14 = 24 marks)

  1. Determine the dimension of the subspace of R4 generated by the set of

4-tuples

f(1; 2; 1; 2); (2; 4; 3; 5); (3; 6; 4; 9); (1; 2; 4; 3)g

(6 marks)

  1. The code words

u1 = 1010010; u2 = 1100001; u3 = 0101000; u4 = 0010100

form a basis for a (7; 4) linear binary code.

(a) Write down a generator matrix for this code.

(b) Construct code words for the messages 1001 and 0101.

(c) Write down the parity check matrix for this code.

(d) Find the syndromes for the received words

1110011; 1001010; 0001101; 1101010

(4 + 4 + 4 + 8 = 20 marks)

 

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