Question

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6) [15 marks] Let V be the vector space of all 2x2 matrices over R. Let W, be the subspace consisting of matrices A such that , + Ay = 0, and W, be the subspace consisting of all matrices B such that B2+ Bx = 0. i. [5 marks] Find a basis for W; ii. (5 marks] Find a basis for W,; iii. [5 marks] Find dimW,, dimW,, dim(W+W,) and dim(W, nw).

Answer 1

Let V be the vector space of all 2x2 R. matrizes eves Let W, be the sub.. spaces consisting of matrices A such that Allt A22 = 0 and We be the subspace consisting 여 matrices B such that Biz + B₂ = 0 (i) Find basis for W, Let the general 2X2 matrla is

AX= A12 A1) A21 A22 Given Alltag = 0. A is zero. e) Trace of il Trace of A sum of diagional elements. trace of A . All + Aqr. Ty (A) = 0 - Ax [:]P. To A The charactic tis equation is. CA(0) - de - tr(A)x+ /Al. - 0² - ox-1=0. x-1=0 7..x = 1! The Eigen values are

If d=1 Thun ( Adi) x = 18.07. (A-) * L:)-(). 0.91, - 212 =0.9, = 0 1.). If A=-1 then (A+11) 9 x T-11 2 N &0. X₂=0. 191,.=0; Ve V2 = LOJ. is The basis of Wi ], 0]

Find the Basis for We Given B matrice such that Biz & Bar B = B Biz B2B,2). Bu – B.21. e B- :) The charaterictics equation is tr(B)n + Bl. CB(1) = x² 2 - 2 - 0 2 +1 = 0. x+1=o. x = -1 + x = 18 ¿,-i The Eigen values are If ai then (A-13) *

R, - iR, 1 17:/97-621 400 :: 21.10. P = R + h) -9; +in, -0. n = -1₂ 2,= in. Vi - --- If t= - i thien 14 +17)x-[; TX=1:1 U-18) PING

R₂R, IR -n,-in, so -1 ( 2,482)=0. 9, +in=0. Lin, X, v;-) over The basis of for W. R. the basis of for We over کم zett but C (IR). is vi-li). dim w, dim W2, dimCW, +W₂.) and ( Find dim (w, nW2) Let A dimwi PIA) # Hlefcas. FIA) is Rank of A non-Zelo rows ( column limon

dimw =PIA RAD. - 2 + 0. dimin, dimin, i dim wa 3.= Li. PIB) = 2, 2(6)=0. dim IN2 P(B) & ad 2 to =2 dim lN2 dimina w dim fw, tlws)? 1 0 B = A 0 (10) 13:11 A TB - 4

-- Rq. 7 R, R₂ P (A+B) = 1 . dim (In, +1N2) = 1. (iv) dim (w, Wz)? 1. :) A = since Wg [Q,;] 22;; = 0,1-1,...} 4-2 = 2. dim (W) mn-m. w-fl'as] | 3 app =0,3-0,2? dim cwo) =mnn. = 4 - 2 = 2 dim (w, AW2) -mn-m-ntl. since m= 2.

dim (WinW2) - 4 - 2 - 2 +1. 4 - Atl dim (WinW2) = 1..