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

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Math 205B Quiz 03 page 1 01/29/2010 Name__ Sat ton 1. Let u, v, 2, b and k be as in the following figure. All these vectors belong to R? and all are drawn in “standard position”. MeTE WELL: mall Lire Y= [2) Re =[2) or any cud, thin. Hus no Stoke gwen, X-ANs ar y- A 1A. Find all ways to express b as a linear combination of u g tks mh =— that le: 2->/ 1B. Find all solutions of zu + av = k. Siniluly 4 K> If - Ih? 1C. Find one way to write z as a LC of u v. = => = Z=- U-3v 1D. Use your answer to 1C to find all solutions of z;u+22v +23z = b in parametric vector form x = p+ v», where p is a particular solution of xu +x2v +z = b and vy, represents all solutions of the homogeneous equation x; u+22Vv + 23z = 0. (Take xg as free). (Hint: From 1A you know a particular solution au + Bv = b. You can add z(z — z) to the left side of this because it would be “adding nothing”. Now rewrite the second z as in 1C and simplify). intended solution : Hit-d= (t2-¥) + x,(Z-2) +6 ) 2 pie 2-49 = (shee B-h-W, (+i 7) + x2 - %( ¥) ne & 42 (4-x;)é +(-I +3x,) ¥ +x,2 -t ( Leip 4 teZ, 2) = 4e x -i Ko xX q % ‘ q é) 6) So X= -1+3x, => X= x2} = |-t +%3 3 x. x, © free NOTE: This quiz continues on the other side. a I Mer nade fo hy fas on py <_<