Question

Let V = M2x2 be the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication. Which of the following subsets form a subspace of V? The subset of upper triangular matrices. The subset of all matrices 0b The subset of invertible matrices. The subset of symmetric matrices. Question 6 The set S = {V1, V2,v;} where vi = (-1,1,1), v2 = (1,-1,1), V3 = (1,1,-1) is a basis for R3. The vector w whose coordinate vector relative to this basis is (w)s = (3,-2,4) is w=

Answer 1

Va M2 42 (IR). net wi- upper fromgulas matain of order 2023 - { [an 2122] : aier } ret [ о 222] tw, B bu [bu biz b22 tWz Now leßo an a12 1+1 bu bizi 0 bz2 922 ant by In al2tb12 9227b22 E WR ie a. ß t W 7 xtß E. Wz 122]tw, all a42 det e EIR and 2- 0922 an a12 cla Call c Call 2 a22 EW, O Cazz 은 it etir; LtW, 27 extW, From 6 and 2 wi Subspace of M2 x 2 (IR) we get is

xet xa Ba Now appa [al + b) We = {] [006] + Manz (IR) an EIR] [01] + Wz [a b] E W2 182] tangan 6,7 & W2 So Wz is not subspace of Man (IB). At Max2 CK): deta Fo} I set of all invertible matsing 2 Wz² A. BE W3 Then Now let det (A) PO dete A #O. det (ATB) deta) +det(B) example + Let A= [8] B[o det (A)= 1812 180 det (B) 162121 +0. but [68]+[!]-[60 we det (ATB) 20 So A B E W 3 * ATB E W₂ There fore W3 is not Subspace of MzxeCiR). 2 ATB 2

Wq n det Subspace of M2x2(IR). Set of an Symmetric matrix) A E Mex_(UR). . Acta A 13. A, BE Wq then Ata A, Beta B. How AtBit Ate Bet AIB. There fore A. BE W4 » ATB E W4 det CERT There (CA) Therefore C, ER, AfWq => CA E W₂ @ From 7 and ② we get Wq is Subspace of Max, (IR). 1 St and 4th option c. Art O c. A CA ::: are

Geinen that Sa{N V₂ V3} be a basis of V = (-lu) IR? where (ا را-,1) د (۷ and V = (1,-1) [W]s- (3,- 2,4). Then wo 3 (~) - 2 (P) +4 (²3) 23(-441) -2(1,-1,1) +4(1-1-1) (-3-274, 3+2+4, 3-2-4) 2 (-1,9,-3) Therefore wa (7,8], [-3]). 2