Question

#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?

Answer 1

Solutions GRAM SCHMIDT ORTHOGONALIZATION. PART-a Process: ③ convert the independent vectors to orthogonal vectors 6 Now Normalize the orthogonal . Vectors Givew 닝 디 [2 ] To The vector first orthogonal can be & itself to, are p= 1670). V A = v= [121 The second orthogonal Vector BaŲ P where is the projection of your P = 14 P- u PAL VSV 2

LIZATION: Normalising these vectors rs LA11 Pokao 2.13 -213t B . B - Hall -self 12²+(-1)74 23 tonat 2 /3 Hence Ortho normal Basis for wa - of our 2/37 -13 213 43 213 / Aus- 1 PART-B You can find the projection matrix m of the Orthogonal Projection PoR W with the vectors Normalised as coluus j.e. kets =5-13 2/3 7 12/3 / 1213 23 - 11 the orthogonal that we will have projection manip M= S(STS)! ST NN

T Now S1-1² | 213 12/3 2/3 -13 2/3 To find az s (STS) IS' STS= -1/3 2/3 2/3 12/3-1/3 2/3 -² 2/3 +243 2/3 ] -Y 2437 (73):68148 (23) (Ha) 4:18 (-3)2+(2/3)* + (243)2 (23)(-) +113)(2/3)+(243)* [10] >> (5") = 1.: eſ lo (Reverse tiend Coo Ruverse Now Tidl CT - Tlo -/2 2/3 2/ T 5-13 2/3 2/37 | 23 - 2/3 | 5 s (sts)'S' za T -73 274 L2/ 2 / 4 / 23 1 (-V)2 + (243)2 )(*2)+(**)(W) (-1*** 21%) 3) (23)+(13)2 f13)**} L - 2/9+419 419 249 419 Islg -4 2/ 7 1-419 519 219 L 219 219 819 1- А