Question

A): compute projw j if u1=[-7,1,4] u2=[-1,1,-2],w=span{u1,u2}. (u1 and u2 are orthogonal)

B): let u1=[1,1,1], u2=1/3 *[1,1,-2] and w=span{u1,u2}. Construct an orthonormal basis for w.

Answer 1

A). We have u₁. u₂ = 7*(-1)+1*1+4*(-2) = -7+1-8 = -14. Hence u₁ and u₂ are not orthogonal. Further, the vector whose projection on W is to be computed has not been described. If v = (a,b,c) is the vector whose projection on W is to be computed, then proj _W (v) = proj _u1 (v)+ proj _u2 (v) = [(v.u₁)/(u₁.u₁)]u₁ +[(v.u₂)/(u₂.u₂)]u₂ = [ (7a+b+4c)/(49+1+16)] (7,1,4)+ [(-a+b-2c)/(1+1+4)] (-1,1,-2) = [(7a+b+4c)/66] (7,1,4) + [(-a+b-2c)/6](-1,1,-2)=(49a+7b+28c)/66, (7a+b+4c)/66, (28a+4b+16c)/66)+((a-b+2c)/6, (-a+b-2c)/6, (2a-2b+4c)/6) = ( (30a-2cb+25c)/33,(-2a+6b-9c)/33,(25a-9b+30c)/33).

You have to plugin the values of a, b, c.

B).We have u₁= (1,1,1), u₂= (1/3,1/3,-2/3) and W=span{u₁,u₂}. Since u₁.u₂ = 1/3+1/3-2/3 = 0, hence u₁,u₂ are orthogonal to each other.