Please PROVIDE EXPLANATIONS 1. Consider two square matrices A;B. Show the follow
ID: 2983781 • Letter: P
Question
Please PROVIDE EXPLANATIONS
1. Consider two square matrices A;B. Show the following:
rank(AB) <= rank(A)
rank(AB) <= rank(B)
HINT: consider the rowspace and columnspace of AB
2.Consider matrices, A belongs to R^(3*2)
of the following form:
A =
a b
b c
d a
(a) Determine a basis for for R^(3*2)
.
(b) Show that all matrices of the form above are a subspace of R^(3*2)
.
(c) Determine a basis for this subspace.
(d) What is the dimension of this subspace?
3.Consider polynomials in P2, i.e. polynomials of degree less than or equal to 2.
(a) Express v(t) as a linear combination of pi(t); i = 1;2;3, where
v(t) = 3t^2 + 5t- 5
p1(t) = t^2 + 2t + 1
p2(t) = 2t^2 + 5t + 4
p3(t) = t^2 + 3t + 6
(b) Do the pi(t) form a basis for P2? Provide a counter-example or prove.
4. Consider polynomials in P, i.e. polynomials of nite degree. Determine whether the subsets, W
dened below are subspaces of P.
(a) W consists of all polynomials with integer coecients.
(b) W consists of all polynomials with degree >= 6 and the zero polynomial.
(c) W consists of all polynomials with even degree .e.g t^2; t^32; t^100 etc.
5. Determine whether the two vector space elements below are linearly independent.
(a) u(t) = 2t^2 + 4t- 3, v(t) = 4t^2 + 8t - 6
(b) u(t) = 2t^2 -3t + 4, v(t) = 4t^2 -3t + 2
(c)
U =
1 1 1
2 2 2
V =
2 2 2
3 3 3
6. Secret Messages
You are endeavoring to design a secret communication system. You have your desired receiver,
but there is also an eavesdropper. For any message you send, m,we have the following received
signals:
r = Hm desired receiver
Explanation / Answer
1)Looking at how matrix multiplication is defined, each column of AB is a linear combination of the rows of A, and each row of AB is a linear combination of the columns of B. Thus the colspace of AB has dimension <= rank A, and the rowspace of AB has dimension <= rank B. Putting these together proves that rank (AB) =< min {rank (A), rank (b)}, since rank(AB) = dim(colspace(AB)) = dim(rowspace(AB)).