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2010, Sec. 1 Practice Bonus
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1. (+5) Consider
Let r = rank(A) and
• 1 = j(1) j(2) ... j(r) = m the indexes of the leading variables of rref(A)
• 1 = k(1) k(2) ... k(m-r) = m the indexes of the free variables of rref(A). We know that for each l = 1,2,...,(m - r), there exists a unique solution of,
.
Let us define for l = 1,2,...,(m - r) the vector ~vl ?Rn such that the jth component is
given by,
? l
-x
?? i
~vl •~ej = 1 ??0 if j = j(i) (index of a leading variable) if j = k(l) (index of the lth free variable) otherwise (any other free variable)
Show that ~v1,~v2,...,~vm-r is a basis of ker(A).
2. Let n ?N and,
V = spanB, B = (1,cosh(t),sinh(t),cosh(2t),sinh(2t),...,cosh(nt),sinh(nt)).
Let D : V ? V such that
(i) (+1) Show that B is a basis of
(ii) (+2) Determine the matrix of D with respect to B.
(iii) (+2) Determine a basis U such that the matrix of D with respect to U is diagonal.
3. Infinite dimensional linear spaces might be particularly surprising. Some of the properties that we already learned for finite dimensional spaces do not hold for them.
In the following let R|N| be the space of sequences of real numbers:
(i) (+1) Give an example of T : R|N| ?R|N| linear such that T is injective but not surjective.
(ii) (+1) Give an example of T : R|N| ?R|N| linear such that T is surjective but not injective.
(iii) (+1) Give an example of an infinite dimensional subspace of R|N| which is not R|N|.
(iv) (+1) Let for i ?N
~ei = (0,...,0, 1 ,0,...) ? V
|{z} ith
be the infinite sequence for which the ith element is one and the rest are zero (analogous to the canonical basis). Is span{~ei}i?N = R|N|?
(v) (+1) Let V ?R|N| be the subspace of sequences that converge to a finite number.
Give an example of T : V ?R linear such that for all i ?N,
T(~ei) = 0,
however T is not the trivial transformation. In other words, there exists some sequence ~v ? V such that T(~v) 6= 0.
4. Consider the linear space Pn of polynomials of degree smaller than or equal to n with the inner product given by,
The Legendre polynomial Lk is defined as the kth derivative of (t2 - 1)k.
(i) (+1) Prove that U = (L0,L1,L2,...,Ln) is an orthogonal basis of Pn.
(ii) (+2) Compute kLkk for each k = 0,1,2,...,n.
(iii) (+2) What is the orthonormal basis of Pn obtained by applying the Gram-Schmidt process to the canonical basis B = (1,t,t2,...,tn) of Pn?
5. (+10) Let A ?Rn×n be a symmetric matrix. Show that there exists an orthonormal basis B = (~v1,...,~vn) of Rn such that . Show how to compute B if