Midterm 2 - Solutions

Midterm 2 - Solutions - 1. Let T : V ——> W and S : W...

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Unformatted text preview: 1. Let T : V ——> W and S : W —> Z be linear transformations. Prove that S T : To if and only if R(T) Q N (Recall that To denotes the “zero map”: Tg($) = 0 for all :1: E V.) \ I éafirfw~ ~~€ (ml w 2. Let V and W be n—dirnensional vector spaces, and let T : V —> W be a linear transformation. Let {1J17 . . . ,vn} be a basis for V. Prove that T is an isomorphism if and only if {T(v1), . . . ,T(vn)} is a basis for W. (ZS-1?} A512; (AM if m ifiié’i‘r’iéfif 6/7246 T [:5 /y*‘ Moi Iii/H] (I 7 Vi I? of 0 n diff (ll/3a W /i {WM ,» die? a (iiicgfmgf (“WW a 5* 9 r 5 g I "(f/1" sf“: “g; i - D I ' a k 4n fix? “KM liftiiflf imam W6 W/ ie «W “W W: “i 2“! k A M j; i K, , ,, W - n V I; if}; NM ; ifs” (iv {mew/i x/siifi. **"‘“’ ‘ j an I; .. J. £4 [3' Aéwmé 5?; 77%} i3} ‘3" fl mg”: é” W“ 7: iii/3M? + m6; }/ «f: TQM; {Iii is; c 2T ' NT) :- W, sit/L if is i f 1*” N9 «2% din/Vi (Wg MM Dimmer T/ T /‘5 4M / “// my"? a (I / 17/1/43? wt 40. iéfi/hfififliégéém, $1 3. (a) Write down a formula for a linear map T : P2(R) —% R3 such that Z (3) 1: _2)7 T(X2 +X) = (1, —2, 1), and T(X2 + X + 1) z (—3, 6, —3). (Your answer Shguld be 1711‘ the form T(a + bX + CX2) = . . . L51 {2.1131114 is» 1“ 45mm M 131.51%, {a My? 1’21 j 112(‘21/1 112% 7.451651 tza/ 5+1: :: é, H V I v a, / Hflfi; {2/ 5351/51 5m; {1: W51 i’iééwfi)’: géwéi ) (In 1111/!“ Ma 1 [sz5be (11%»11/1’k W 711% W + 5 X1); W? W 11% M) I 'w ’v ’z‘ j w”? {g \ )1 IV; I / (é: 9,:{jlg1’wgi “’3' {fiwfifg [5"2; {fl -¢j«j) : + “1:13 : 1111:5141}111M ) if M “2M” 34 k (‘0) Compute rank(T) and{31111161711:1‘1371“afiafifipmsm. 0m 11515113: NUS: 5WJA(ET(XZ}, T (:1 111)}; :1 4. Define T : P2(R) —> BUR) by = - X2 + f’. Let [3 2 {X2 — X — 3,X2 +2X+ 1,3X — 2} and let 7 = {X2,X, 1}. (a) Compute the matrix [T]? (13) Let Q be the Change of coordinate matrix that changes fi—coordinates to 7~coordinates. Qompute Q. (M J; (C) Using your answers from parts (a) and (b), write down an expression for the matrix [TM (You do not need to'muitiply out the matrices.) V i E! «WW I WW? gm: MW: 2 3< é i *2 ...
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This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

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Midterm 2 - Solutions - 1. Let T : V ——> W and S : W...

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