Question

Hello, I need help solving this linear algebra problem.

1. Let L be the set of all linear transforms from R³ to R².

(a) Verify that L is a vector space.

(b) Determine the dimension of L and give a basis for L.

Answer 1

1. - 10 1 L = FT: T IR3 (Rt is lineur map Let I,UEL and der (T+u) (x + y) = (a+y) + u(x+y) Tel + Tryl + vajtu yg = 5 x + 1 Tal+ Ult = (T+U) x + (T+U) y (1+0) (xx) = T (dx) + U/dx) - & TX) + & UCKS - (TX) + U ) = 4 (1 tox Ttu is a linear map TUEL (2T) (x + y) = (2T) Cu + &T) (W - & Tat & TUJ = x (Tal+T ) = a Texty) (QT) (B x1 = & T (B) = 2. B T (X - Q TX 10 : - XT EL ( is closed under and scolar multiplication addition

4 (T+ U) (V) = TN + UU to VER Ww & Tui = = (U+T)(w for allvar I Ttu= UIT 0 (T+O) U = Tus+ 0 = Touto L = Tow for all VG (R3 7 T+0=T 3 TutD) (w=TW + (-1) - T + T (-) = 7 (v-v). To=@ for alt ver -> T (T) = 0 For call ut R, 1 (T+U+W) Ws - (T + 0) CJ + wcuj. - TOST U (y + Weus = + (U WA WW) = 6 + +wJ 7 (T+u+ w = T+0+w) (@B) D W = BUMBUR ET (JAV). = a (B.T WIB 22 (81) )

T for all ut (R² VEIRS 7 Copy atRT) 9 (1.T) (W = 1.Tu = T (1.1) - Tcr for all a F-T 0 L. (I + U) () = x (T(v) + U) - DIU + avu = (t + & JEG for all = (+ U VG (R3 2. tu att du Smilarly. (246) J - LT + B. I 7 L 6 a vector spus chin (L) = dini (k3). dim (18²) = 3x2 = 6 toel Toro Tool 0-100 Moo The S inafar Let TEL Then I (R3 (R2 is linear treens :

T = A where = { liger, egy is standard basis of R3 and r = 40,0), (011)} IT) is 3x3 matinx- Je Mexa Cor). Since, s is buses of M2x2 (IR). where I (3) = (x,0) To lů) = (2,0) T5 () = (0,4)

het TE l be any linear map * ProF: 01) taan (T) + as [is] p + de 178 - I G Span & Tin а ; Т. 2 р T т 73 km2 2 4:20 { Tig o 3 is basis of L