University of California Los Angeles Linearly Independent Subset Discussion

1. Let T be a linear operator on a finite dimensional vector space V .

(a) Let k be a positive integer. Prove that R(Tk+1) ? R(Tk). Conclude rank(Tk+1) ? rank(Tk).

(b) Prove that if there is some positive integer k such that R(Tk+1) = R(Tk), then R(Tm) = R(Tm+1) forall m ? k.

(c) Suppose dim(V ) = 3. Prove R(T4) = R(T5).

2. Consider the vector space V = R4 and the linearly independent subset

S = {(1, 1, 0, 0),(0, 2, 0, 2),(0, 0, 0, 1)}.

(a) Apply the Gram-Schmidt Orthogonalization Process to get an orthogonal basis ?0for span(S). (Youdo not need to prove that S is linearly independent.)

(b) Use what you found in (a) to find an orthonormal basis ? for span(S).

(c) The vector x = (1, ?1, 0, 0) is an element of span(S). Compute [x]?.

3. (a) Let F be a field and let A, B ? Mn×n(F) such that A = P?1BP. Use mathematical induction to provethat Am = P?1BmP for all positive integers m.

(b) Suppose A ? M3×3(R) is a diagonalizable matrix with characteristic polynomialpA(t) = ?(2 ? t)(3 ? t)(1 + t).Compute the eigenvalues of A4 and then prove that A4is diagonalizable.