(b) (3%) Let X and Y be two subspaces of V. (i) Write the definition of XtY . (ii) Write the definition of X⊕Y . (ii) What is the mathematical relationship between dim (X⊕Y) and dim(X⊕Y)?

(c) (2%) (i) What is the condition on R(A) or N(A) that is equivalent to the existence of solution to Ax= b? (ii) What is the condition on R(A) or N(A) that is equivalent to the uniqueness of solution to Ax = b, if it is solvable?

(d) (2+4%) When the equation Ax = b is unsolvable, we may consider the so-called least squares problem to find a set of solutions, having the least squares error, from solving a normal equation. (i) Use the property and condition mentioned in (b)-(c) to explain why the normal equation is always solvable.

(ii) Suppose that rank(A) = k ＜ min(m, n) and let A = BC be a full rank decomposition of A . Use the known matrices B, C, and b to describe the unique projection vector p of b onto R(A) with the least∥b-p∥2.

(a) (1+2%) Let δ be a unit element of"t", the addition operation of V. (i) Write the equality condition about & as shown in the corresponding axiom. (ii) Let y be another unit element of "+". Show that δ=y.(須註明使用到的所有向量空間定義中的公設,否則不計分)

(c) (2%) Let 〈●，●) be an inner product defined on V. Show that the function defined satisfies the triangular inequality property.

(d) (2+3%) Consider the vector space R2✖2 with the inner product (A,B):= trace(ATB) and denote Y:= .(i) Describe Y⊥ as the span of an orthonormal basis. (ii) What is the matrix, denoted by PY⊥, that represents the orthogonal projecting operation ⅡY⊥ : R2✖2 → Y⊥ along with the subspace Y with respect to the ordered basis E = {G,H}, where G and H are vectors of the standard bases for Y and Y⊥, respectively?

(a) (1+2%) (i) Write the definition of superposition principle and (ii) show the implication from the combination of additive and homogeneous properties to the superposition-principle.

(b) (2%) Write (i) the definition of Ker(L), the kernel of L, and (ii) the definition of L(V), the image of vector space Y, respectively. Ib-plz

(c) (1+2%) Suppose L is linear.(i) Use Ker(L) to describe a mathematic condition that is equivalent to L being one-to-one. (ii) Moreover, show the implication from the condition on Ker(I) to L being one- to-one.

(d) (2+4%) Let L be linear with A as its matrix representation with respective to bases E = [V1,...,Vn] and F =[w1,. ,wm] for V and W, respectively. (i) Describe a mathematic condition about A that is equivalent to L being onto. (ii) Moreover, show the implication from that condition to L being onto.