15 21. Let = , u₁ = = , uz = (a) Are u and 2 orthogonal? 3 and let W be the subspace of R4 spanned by u₁ and 2. (b) Decompose as =₁₂, where ₁ is in W and 2 is in W+. (c) Which vector in this problem represents the closest point to in the subspace W? (d) Find the (minimal) distance from y to W.

Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.

Find a basis for R2 that includes the vector (2,2).

Consider the vectors u=(6,2,4) and v=(1,2,0) from Example 10. Without using Theorem 5.9, show that among all the scalar multiples cv of the vector v, the projection of u onto v is the closest to u that is, show that d(u,projvu) is a minimum.

Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.