Question

3. (a) Show the set of all matrices of the form х A у x + y + z 2 is a subspace of the vectors space M2(R) of all 2 x 2 matrices with entries in R. (b) Find a basis for this subsace and prove that it is a basis. (c) What is the dimension of this subspace?

Answer 1

Soi & W= Let v= M₂ CR) be a vector space x У A = € M₂ CR x, y, 2 ER x+y+z Now x=0=y=2 ER Then EW Wis non-empty subset of V-M2CR). x2 2 Now A= - & B = Z i xty, +2, Zz X + Y₂+Z2 Let for any scular A, B EW & & be any Sie x +GEIR, LGER then x, 4, 2,, 02, Y, Z, ER. & XER Y, TYRER, LY, ER У, 2, +Z2 ER, 22, ER x A+B= + 2, x +4, +2, x + y2 +22 aq 22 x+x2 4, +42 - 2, +Z2 (24+4, +2, + OlgtY2 + 2q) G+X2 4, +42 2, +22 (24+x2)+(9, +42) +(2,+22) y A+B= z x+y+2 where x=x+X₂ EIR y = 4, tY2EIR f 2 = 2, + Z2 EIR A+B Ekl. Wis closed under addition, 3G 204 X A = { 2 x4 +4, +2, 22, 264+4, +2,) LA = 204 221 LY 204 + xy +22,

у where x = 26 ER LA= Z x+y tz y = 84, ER 2=22, ER XA Ekl. . Wis closed under scalar multiplication. У 2 x+y+z His subspace of V-M₂CIR) ie the set of all Matrices of the form A= is subspace of vector space M₂ CR of all 2x2 Matrices. with entries in IR. y Let W = A = E MECIR) |X, Y, ZER x+y+2 2 D. z X, Y, ZER {"l! + y - o 1 W. { 1, V, the Ve thg Voz / 0x = 4, y = 22, 2=43 v=[:] x= [0 ] - [3 :) WELCB) wheve B = { V, VZ, vg} f V, V2, V3 € WEV where ie B spans W. consider. L, V, + 2 V2 + 2 V₂ = 0 [ ]+[:]tes[:]-[:] 1 L2 0 0 O di 1, the th3 comparing the matrices, we get L, = 0 = 42 = L & s, the th=0. i. x = 0 = x2 = 43 i.e there exist Scalarst, L2 & 3 all are zeso such that L, V, 22 V2 + z = 0

B is linearly independent subset of kl. B = {v,, V2, vg } is a basis of ul. where w;= [0] =[ ] * * = 10:1 NOW B contains three vectors dimch) = numbers of vectory in B = 3 dim che = 3 The dimension of this subspace is 3