midterm 1 fall 99 solutions

# Linear Algebra with Applications

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Unformatted text preview: First Midterm Examination Your Section (circle one): No calculators are allowed. Last Name: First Name: Mathematics 21b October 25, 1999 John Boller Noam Elkies MWF 10 MWF 11 Score | Ilﬂlllﬂlﬂllm IIEIIIEIHIIH IIIEIHIEIIIIH lllﬂlﬂlEIIIll llﬁﬂﬂllﬂlﬂllﬂ SOLE/TI 0N3 1. Consider the system of linear equations 3:33 — 2x2 = -3, \$3 +2334 +4231 2 1, 2m1+\$2—x3+m4_=2. (a) (2 pts.) Write the coefﬁcient matrix and augmented'matrix for this system. 0 _7_ 3 o o -z 3o§—3 A= 4 o '1 2 [Nb]: Li 0 *1: l 1 l—I ! ‘7. l 4t H2. (‘0) (5 pts.) Calculate the row-reduced echelon form of the augmented matrix. 0-13 Olin-3 1%“ii'l l'ii'“|2 l HOIlllﬂ—34OIZIHO’Z301-‘3 are: 112 o~2 sci—5 0-4 3 rig—g ||I llr‘ 1%":E" lozz't r“? ' '3 Ol—-%_o:%—"olr_§_o;; oooolo 000010 (c) (3 pts.) Find the general solution of the linear system. Verify that your answer does in fact satisfy all the equations. ' i -L Le‘l’ xa=l *. ﬁ-ﬁ-s-Ji‘t 3' —.t z I 3 3 Lil ‘3” 7“ 2 32+?) = 3: +5 2 H. O 0 TL»- 7‘1: "2.4%5 ‘c’ 5 O I ‘ * o in: ink?" ** “k 0 3 s q I-Iv — "—- Ebl 3(5)—2(;+;s) .. 35 2: 35.4 3 1 1 J. _. _. E3 2- “it * ‘i_(q-;s—~2{) .. s+2£+l s 2. Let A and B be the matrices 1 0 0 1 0 0 A = 0 0 0 , B -= 0 0 1 . 0 0 1 _1 0 (a) (4 PW) Describe A, B geometrically as linear transformations. _ I? A??? '5'. BF; 2 ‘3. “i=3. - . '3;- = "E; IN 3 A63 2 3‘3 BE: = a: A 7: P‘thc’h‘“ M'l’vi‘iiu \$2'Fiomt . x B WW“ E CLth7€ 4M I32 «flank (b) (3 pts-.) What are the ranks of A and B? Is either A or B invertible? Justify your answers. m4 (Mace I 2. Shut? rug = [ - «MA A r, m; “out. 000 "Ml Lune: 0 IM 8 ~‘r Mva-i-iisie. (TL madcap-x haka 57 9 R a vain-L»... 51 we.) Whig): 3 I smir mi‘LBi-Z‘ _[i‘>?§1 00‘ (c) (3 pts.) Do A and B commute? Interpret your result geometrically. Ae=[\$2§] BA=[\$ZCI] ABqEBA o —| o . o o 0 (AW: 22‘. (BM 33.: a: (we? is --'é: (we: 0 -—v - -9 (A931: @ O {61536.3 r:Z 33“? (Am: «2 fach - 2 Enayteam= tag-flaw Ker (A3} 2 ?--axis “(310 '—“ 3 mm; (a) (1 Pt. each) Deﬁne: W T'- Rwﬁ‘Rw‘ k a [War wscwmdem u-Prk we be A. am: like“ \‘4‘5?’=3’3 O kernel 0 image I-«(M 2 § aeﬁlml ARE-.5 3v 5m “huff 0 rank Mum; #129th 12 a» meme) a span TL: sip» a-l] 4L; Eel ope-eel”: €73“... F: 5;} or all fossil! [Mm- WLMLM J ,{3 ﬂ.“ vec'l-av: :: g C‘TyT-r-mi cn?” Ch...qu m rulers; 0 basis I V ‘5 a subsrace a-C laymjgm e V, “Hun. §\$§fhlﬁ 1'? IL Lam} pun-V :5: H1 4-7.1“ A” [MI-L1 Md 1):; ,PMV, (b) If A is a 4 x 5 matrix of rank 4, o (1 pt.) Is A invertible? Why? Halon“ 1610* mav'l'Hui 'Mﬁt 1" 'I‘WFLLC ' o (1 pt.) What can you say about ker(A)‘? lemﬁi J. o (1 pt.) What can you say about im(A)? Jaw = Ll o (2 pts.) What can you say about a linear system whose coefﬁcient matrix is A? 11”. Sblvhh at“ (9! a. [ﬂu . T ace of R3 with basis consisting'of the 4. Let T : R3 -—> R3 be orthogonal projection to the subSp 2 single vector [ -—1 X . [Note: this is go; a unit vector!] LA :3 z ' O o ‘3 2:} a .7 u ' a I a — ___.——- h I m QIWTERI FFOJﬁV' w (a) (4 pts.) Calculate T (51, T 62, T €53. .a, 2. = - [it] 6 o A“ 3- l W L [:1 [-11 = a"? = iii “1cm: "EH-’2 j : -L[}[_l : s o 5 0 '5; 7.. is LE1 [it as} m (b) (2 ﬁts.) Find the matrix for T. A: ___;_ a; O O 0 O (T) and im(T)? Find bases for these subspaces. (c) (4 pts) What are the d1 ensmns of ker W5 -7, “xii-£0 Lat-\$2]: m T}; Twi(ﬂ= sf” [000] M‘ 1:25 [Fl]:[ 5] 0 I o O 00 MM: %5 1‘: _€ 4. l ' “W o l Catawba: 7. ...
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