Mathematics

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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

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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.