Problem 4. Suppose V is an n-dimensional complex vector space and TEL(V) is such that dim...
8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T (a) Prove that the dimension of each generalized eigenspace of T is at most (n - 2). (b) Show that (T-1)"-2(T-21)"-"(7-31)"-"(a) = 0V, for all α є V. 8. Suppose V is an n-dimensional complex vector space. Suppose T E C(V) is such that 1,2, and 3 are the only distinct eigenvalues of T...
6. Let V be a n-dimensional vector space and let TEL(V). Which of the following statements is not equivalent to the others? (a) null(T – 2 Id) = {0}. (b) a is an eigenvalue for T. (c) T-2 Id is not injective. (d) T-2 Id is not surjective. (e) T-2 Id is not bijective. (f) T-2 Id is not invertible.
P.3.31 Let V be a complex vector space. Let T : Mn → V be a linear transformation such that T(XY) = T(YX) for all X, Y E Mn. Show that T(A) = (trA)T(In) for all A EM, and dim ker T = n² – 1. Hint: A = (A - tr A)) + tr A)In.
Prob 5. Suppose V is a finite-dimensional complex vector space, T E C(V) is diagonalizable, and all eigenvalues of T are strictly below 1 in absolute value. Given V, and u є V, what is limn oo (Tnt)?
Suppose that T E End(V) where V is some 3-dimensional real vector space. Suppose that det(T) 1 and that i is an eigenvalue of T. Find a, b, and C such that T5 = aT2 + 6T +c.
Suppose V is a finite dimensional inner product space, and dim V = n. If is an orthogonal subset of V, prove that a. W can be extended to an orthogonal basis for V. b. is an orthogonal basis for c.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent. 10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
2 (5 points) Recalled that null space of a matrix A € Mnxn is defined as N(A) = {r € R” : Ar =0}. Now, the eigenspace of A corresponding to the eigenvalue 1 (denoted by Ex(A)) is defined as the nullspace of A-XI, that is, EX(A) = N(A – XI) = {v ER”: (A – XI)v = 0}. You should have three distinct eigenvalues in Problem 1 above. Let say there are li, 12, and 13. (i) Find the...
6. (a) Suppose that Wi and W2 are both four-dimensional subspaces of a vector space V of dimension seven. Explain why W1 n W3 {0 (b) Suppose V is a vector space of dimension 55, and let Wi and W2 be subspaces of V of dimension 36 and 28 respectively. What is the least possible value and the greatest possible value of dim(Wi + W2)?