# SIT292 LINEAR ALGEBRA: Vectors and spaces 2013

# Mathematics Assignment Samples

You can download the solution to the following question for free.

(ExpertAssignmentHelp do not recommend anyone to use this sample as their own work.)

## Question 1:

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

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

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

vT ( 1; 1; 1; 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 ( 1; 1; 1; 1)

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

independent vectors in V .)

3. Find a basis for the subspace S of R3

spanned by

fv1 = (1; 2; 3); v2 = (1; 1; 0); v3 = (2; 2; 3); v4 = (3; 3; 3)g

4. Determine the dimension of the subspace of R4

generated by the set of

4-tuples

f( 1; 1; 0; 1); (0; 1; 1; 0); (1; 1; 2; 1); ( 1; 2; 1; 1)g

State one possible basis for this subspace.

5. The code words

u1 = 1101010; u2 = 0100010; u3 = 1100011; 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

**Solution**

a)

Let w=(-1,1,1,1)

Note that 0V since 0.w=0 so V is non empty.

Let v1 , v2V and c is a scaler.

Now:

(v1+v2).w = v1.w +v2.w = 0 + 0 = 0

and

(cv1).w = c(v1.w) = c0 =0

Hence V is a subspace of R4.

For complete solution please download from the link below.

```
(Some parts of the solution has been blurred due to privacy protection policy)
```

[mc4wp_form]

**Check similar Samples**