SIT292 LINEAR ALGEBRA: Vectors and spaces 2013
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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
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
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
Note that 0V since 0.w=0 so V is non empty.
Let v1 , v2V and c is a scaler.
(v1+v2).w = v1.w +v2.w = 0 + 0 = 0
(cv1).w = c(v1.w) = c0 =0
Hence V is a subspace of R4.
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