Question

For the following six questions, indicate whether the following statements are true or false. In each case give a reason for your answer. Problem 13 [10 pts) If L:V +W is a linear transformation of vector spaces and U CW is a subspace of W, then {v € V | L(v € U} CW is a subspace of V. Problem 14 (10 pts) The set {A E R2x2 | A is nonsingular} is a subspace of R2x2. Problem 15 (10 pts) If A, B are two n x n matrices, then A is similar to B if and only if pa(t) = po(t).

Answer 1

13. is a c w is Let Lii w linear transformation of vector spaces.. Also, subspace of w. Now, let it (u)-{var: 2 (1) au} have to that L-1 (o) is subspace of vo not subspace of v. Since L (o)=0 EUR of Lt (0) show we or So, t*, Now, tet x; y E L-1 (u) Then, L (2) Lry) E U. As is a subspace of w Then L (x) Lry) L(x+y).EU x+y E L-1 cu) So, I'(U) is dosed under addition, Let any scalar and and KEL? (0) Then, L (2) € U AL (2) Eu as vis a subspace L (xx) EU L-1 (u) L-1 (0) also closed under scalar multiplication Therefore, L (U) subspace of v. a be λα E So, is is

14 Let {A ER²X2 A ER²X2 : A is non-singular} subspace of R2x2 identity, the zero Now, V is not The additive As, matrix 0 ¢ V. Also, I - [i] EV & 1 -I = But € v I + (-1) - [:] & So, vector subspace of IR²X2 V is not

A and B are nxn 15. Let matrix. Let A is enists matrix similar to B. Then there invertible nxn P such that The Now of B p-1 AP. characteristic polynomial is PR (t) = det (B- t In) - PB (t) = = det (82 AP - t. p^? InP). det (PAP - pa (t In) .P) = det (p-2 (AP - (t In). P) PB (t) = det (o-t (A-tIn).P) P₂ (t) det (p1). det (A- tIn) det (P) det (A-t In). → → U -> IB (t) and B=[ ! : PA (A) Conversely, let A= ] Then det (A-t Iz) = (x-1) and det (B-t Is) = {n-1². But the identity matrix.is similar itself is not similar. А of this The Then given statement not true. B is to only Soi and converse is