Information - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

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Math 205B&C Quiz 02 page 1 01/30/09 Name. Ste edad Gln Section: 8am 1:10 pm 2 5 1 -6 12 il 105 0 -6| 7 io 6 3G ee (0 L 6-0: 0}—3 LetA=/4 49 15 _8 14 and b=], ; then RREF of [Alb] is 06-0: 4l—5 1 3 § -2 2 2 0000 oO} 0 1A. Use the above information to express all solutions of Ax = b in the form p+ -v, where p is a particular solution of Ax = b and v;, represents all solutions of the corresponding homogeneous equation. we have oe >} 6 € Li + On t0%¢ ele % 5 me Ie (ash) — & Xs] Ole lt fers of, thn x, 8x, Wyre ty 2 ° 4 one free, a ee (x5 6 free) O o ! r ee 1B. In terms of the definition of linearly independent, do the columns of A form a linearly independent set? Explain your answer. s : I> > = Tn 1A, we sum there axe ron. taint altos J x= © which means there ee Solution. 7 ¥,4, 4 xa, + ¥,4 #4, Gee =e D ra 5 - w yet not aah J x, m1, 2,0, Me here be O. "law DERMITION LT 6 shed. 1C. Label the columns of A as a), a2, ..., a5. Show explicitly how to express ay as a linear combination of the first four columns. Give two different ways to do this, one of which involves a non-zero weight for column a3, while the other does not use ag (ie, its weight is 0). (Write your answers using the symbols a, a, ..., as; don’t copy over all those columns of numbers). “pe Sche these podolams, well netet sdutims f Kat HE, = Bin the eg td nen (chi it alos woh “eile fe" Be) Since x, 6 fee | see A] wel clnose x = 2 The 17 Queston tasks toe @ ron gore Wcijhte tor By jp Since Xy 6 tho tree ux hese x,= 2. Te obfae serbia beers x)= “546213 Mn Os tym So [A Oi, old, +46, +28 jin (Ho -% ~ 8 1a ; The 24 Quaton duly for A wesht f Ot 43 @ ort anooe ee... 5 Now x, %@ OF6=6, ¥2=0, x,=0, %y= 4, %5ALy So Ga, +, +4, =0 § 1D. Explain why column ag can not be written as linear combination of the other four columns. 0 ee By solA eojly +%ydy ewe, Le scolert A gty oy The Baaa,-f, 40a, ee a ul we ‘hoe ach f& AR=8 in te ergot &e-l, Tes at inp ldle, 0 OP cower Hhe ectigete Bb On day such colton, @ OR* Since reer ( (4 AAS, |2]) hus [oot] a re lat rs ue cant sabe xf, ya +f, 1E. Do the columns of A span R*? Why or why not. + 4,24, No. ih pelle tea be Som BS Ry PREF (ALE) din [eveovdt) mire leat rev, shai Axe ne an soln, intonss A, dhe