Question

Find the best approximation to z by vectors of the form C7 V + c2V2. 3 1 3 -1 -6 1 z = V2 4 0 -3 3 1 The best approximation to z is . (Simplify your answer.) - 15 - 8 8 - 1 Let y = , and v2 Find the distance from y to the subspace W of R* spanned by V, and vą, given 1 0 1 - 15 3 3 - 13 09 that the closest point to y in W is = ch - 12 The distance is (Simplify your answer. Type an exact answer, using radicals as needed.) 1 5 Let u, - 2. u2 -2, and uz = 1 Note that u, and u are orthogonal. It can be shown that uz is not in the subspace W - 1 1 spanned by u, and up. Use this to construct a nonzero vector v in R3 that is orthogonal to u, and uz. The nonzero vector v=is orthogonal to u, and uz. 3 12 V10 Let y = - *:)". , and W = Span{u}. Complete parts (a) and (b) below. 10 a. Let U be the 2 x 1 matrix whose only column is u. Compute UTU and UUT. UTU-[O] UUT = (Simplify your answers.)

Answer 1

3 from egun un (21 422) we have 2 - [! 5 -2 1 R₂ 25R, (700 [2]U-(0) R21 R2 Y ie X+24-3=0 24-320 29=3=k No. of t. I solution = = na f(A) choose rastitorasy) 3-2 - 1 x- is vs (i) - ky kli (324k =Yzk, 3=244 ( (choose K21) Ang obviously <u>20 & CN, 4270 (iv) Here y-VE) -13 / & W = span{ up} 30 = i lo lax) UTV a too [3] 13,-3. 12)- to [ 9+1) =} (10) = [!] ) OUT theo [ 27 13,-1)=5+ [2]-T (6) UUT 3 To öt oliv

3 I solution (0) 3 -6 Here Z? = 14 -3 3 3 "2 = gu tala = I q +₂ ㅗ -3 -2 3 = yam 34 to -ą ta .-3c2 9-2₂ which gires 34+C2=3 -G+q=-6 -3ů =4 4-26=3 2= ¢ 3 aq tq=-6 7-9 = 6-2 =-6+1 3-42-1844 -12 3 a 3 3 Hence from (2), Z=14 het 2/3 (74, -202) 14 + (4) V2 Cave-22) Dr Za Ang (11) Here 15 I VIS (l -1 I -15 3 ::6, a ve (1,~1,-1,3) را ره۱۱ (در احرام ما ) 1+1+1+9 (-8,1,2,3) es V2 111211 (-0,1,0,3) 64+1+0+9 va se

Now we calculate, Pry), Poly) <y,exes +S4, e> ę <<-15,4,4,-15), (),1,-1,3), (1,-1,-1,3) VT V12 3(-15.-1, 1-15),(-8,1,0,3), (-8,1,9,3) V74 94 -15 +141-45 5(1,7,-1,3) 1204+0-456-3,4.93) 12 74 -5(1,-1,-1,3) + 74 (-8,1,2,3) 74 Poly) (-13,6,5,-2) Therefore the distance is d (4W) lly - PocY311 ll (-15,-), ), -15) -(-13,6,5-37|| = Ell (-2,-7,-4,-3) || V629²+1-78+ (-4)2+(-3)2 =√78 dCY, W) = 178 Ans (11) Since Waspan{us, uz} (uz is lincerly ("Uz is lincerly independent) bet Na JER? such that ३ (vų >=0 2(x,7,3),(1,2-))>=0 atry - 320 and <N, U₂>=0 iie <<«,4,3), (5,-2,1)) 20 5x2 -28+2=0 on