Question

Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.

Answer 1

(۱٫۱ را را-) = 4 (ارا را- ۱) { = ا :: )+(!)*(4)(!) Mu/۱ - ): ): ):( | = ال۱۱ = 11,ا|1 (54) ;({}() :()({ ;({}( دهار، با = با = 6 = < <,, ,بار {u, 42} is not an orthonarmal set. So ew (5ر5 ,, - = با 6 (۱۱ ۱۰ را) + (1 را را- (۱): (و را را راح ليه هامي Xlor ي امي Xl 2- =f - له د T 10 م د » , با هد -( رو (۱ رای مسا) (ا را را ر۱-) + (۱٫۱ راما )

© To find the orthogonal projection, we first find the arthogonal complement of W. lut cu,y,z,w) E WF. then <cury, z, w) C1,-1,1,1)>=0 and <cuy, 2,w), + (-11,1)>=0 U-4+2 +w -O -+y +2 +W=0 by addig there we get by subtracting these we get Z+WO x-y=0 T2=-6 This cu,y,z, W), Can be claasen as (1,1,0,0) and (0,0,1,-1) Hence {(1,1,0,0), (0,0, 1,-1)} will form the basis of wt. now E1,1,3,5) w,16 +1,1,1) + uz (+1,1,1,1) + uz (1,1,0,0){Wy (9917 4,-4₂ +243 = -2 + => => -+42 +43 = 1 ) 2,-42-243 = -2 adding these two we get 22, 24, =-4 Th1-42 = -2 - also u +42 + 2y =3 = 2, +42 +244 = 6 W+ U2 - 4y = 5 ) nitu2-244=10 adding these two we get I mitu, = 8

- ا ا ا \\ - = ر- +%2 = 6 Solving these two we get ( 23 ) =5] This projectionſ is آ = (5ر3 رار ا 5 + (1 را ر-ر1) 3 . 1 (۱٫۱ را را - 3 + 5, 3+ 5, 3+5) ما ربا را را- ع - (5ر3 را را-۲