試題詳解

19. Which of the following statements are true?

 
(A) Given A and b of proper dimensions, when b ∉ R( A ) the linear equation Ax = b has no solution. However, the associated LSP (least squares problem) is always solvable and the solution is unique.

(B) Let V be a vector space such that V = X ⊕ Y. Then only when X ⊥ Y, i.e. the two subspaces are orthogonal, can two projection mappings, say P: V → X and Q: V → Y, be defined with the complementary property P + Q = I, where I indicates the identity mapping on V. 

(C) (continue from ( B )) The projections P and Q become orthogonal projections when X ⊥ Y. Moreover, once the bases for X and Y are chosen, their union forms a basis for V, and the two projection matrices associated with projections P and Q with respect to this set of bases are all symmetric.

 
(D) Any projection mapping is a linear transformation that is definitely onto, but may not be one- to-one.

 
(E) Consider the vector space C[-1,1] with an inner product (f, g) := f(x)g(x)dx. Then the set {u₁, u₂} with u₁ = 1/√2 and u₂ = (√6/2)x forms an orthonormal set in C[-1,1]. Moreover, the best least squares approximation to h(x) = x² by a linear function is = (√2/3)+(√6/4)x.