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Math 205B&C Quiz 03 page 1 02/06/09 | Name Suscache solton Section: 8am 1:10 pm . 152 + 10x» + 3523 1 1. Suppose T : R? — R‘% is the linear transformation defined by T ([#!) = st : = pa a 5 1 2 3 a 102 + 8x2 + 1823 la. What is the value of p? @ ...and the value of g? @) 2 2 30 +0435 os 1b. Find the image under T of |0|. 7 ON 32+0 +24/= SG i : 24 T4209 44 ZOoto+ lg B28 1c. Find the associated, or standard, matrix A of T. Is 10 3g A _ lhe mM 24 = Wz fo 20 by 10 & 1g b 1d. Use A in a superaugmented matrix to find explicit conditions on b = ”2) that will guarantee that Hite be bs by bs iS 70 3¢|] oe bo s loo 24 |b bis inthe range of T. [46 m 2]o | col wr}O ' -4/0 © 52-3 12 10 20]6 ef o 00 6]1 oo ¢ ~I5/2. 10 8 I8]e oot © 0 O}]o 1 -3 2 The list tue rous Shot T(X)=6, o AR =b 4 has @ colotion E> So +b, 5b, ~ "by by O- be ~3b, +26, le. Use (1d) to find a value for each of b; and b2 so that d = 1 is the image of some vector x in R?. wit, L=1. ane b,=2 in the above tr» coneiitons, | 2 we And b,=/0 and b,=-| 1f. Let d bees as in the previ ious problem, ‘and find at least one x for which T(x : =d. by bain 71x) = tha @ salan ge ee (2b, +h, ‘Sx, 22> x,=(2#5)- 5x; KF Reo: X3 is x (Abs ~3h + Ix, Xe (A- -6)+ x “We Yy. 1g. Is T onto R?? Explain i a = of the definition or any equivalent statement. about A | X= Ay No, Zo id nese a 73) ee ash act: eH ne cOUBIN Big | gg, J (or in tems dA, RREE(A) hus rots J OF, A =8 ttm yield bance job ff dt anh") X50 we gut 1g. Is T one-to- Get Explain in terms of the definition or any equivalent statement about A. yale 3 s|e Ne & Ps a Pel) 18) komt 29) 9 dL. al if x,=L we or, R)=b DRS since PB hwo tree vrnbles (so AR=E hor co-moey schtins) 2. Now let T: i —+ R? be defined by T (é }) = P* = | 2 LQ 2A. Find the following: T(e1) -7[) -ohg T(e2) = af }) [2] T(e: + e2) = 7 [‘): lee 2B. Use the results from (2A) to explain why T is not a linear transformation. pide thet Tl2)+T (2) =[3]+[']=[ 2] dae TE) = [25s i] 4] me hae Te Tea so te Tlit) +72) # Thad) toe ol GER’, 4 vaton Hyp Mi el A here eee 
