課程名稱︰線性代數二
課程性質︰數學系大一必修
課程教師︰謝銘倫
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/04/20
考試時限（分鐘）：110
試題 :
Part A
Problem 1 (10 pts). Let
2 3
V = {a + a x + a x + a x | a , a , a , a ∈ |R}
0 1 2 3 0 1 2 3
be the vector space of real polynomials of degree ≦ 3. Let 〈, 〉 be the inner
product on V given by
1
〈f, g〉 = ∫ f(t)g(t)dt, f, g ∈ V.
0
Let W = |R．1 be the one-dimensional vector spanned by constant functions. Find
⊥
an orthonormal basis of W .
4
Problem 2 (10 pts). We equip |R with the standard inner product and let V be
4
the subspace of |R given by
4
V = {(x , x , x , x ) ∈ |R | x = x + x , x = x - x }.
1 2 3 4 1 3 4 2 3 4
Let v = (1, 2, -1, 1). Find the orthogonal projection of v onto V.
Problem 3 (25 pts). Let
╭ 1 1 -1 ╮
A = │ │.
╰ 1 1 -1 ╯
(1) (15 pts) Find the singular value decomposition (S.V.D.) of A.
(2) (5 pts) Find the Moore-Penrose inverse of A.
(3) (5 pts) Let W = {(x , x , x )| x + x - x = 0} be a subspace of the real
1 2 3 1 2 3
3
vector space |R with the standard inner product space. Find the dimension and a
⊥
basis of W .
Problem 4 (25 pts). Let
╭ 8 2 -2 ╮
│ │
A = │ 2 5 4 │.
│ │
╰-2 4 5 ╯
t t
(1) (15 pts) Find P ∈ M (|R) such that P P = I and P AP is diagonal.
3 3
(2) (10 pts) Suppose that B ∈ M (|R) and C ∈ M (|R) such that BC = A.
3 ×2 3 ×2
Find CB.
Part B
Problem 5 (15 pts). Let (V, 〈, 〉) be a finite dimensional inner product space
over €. Let T: V → V be a linear transformation. Show that T is self-adjoint
if and only if for every v ∈ V, 〈T(v), v〉 is a real number.
Problem 6 (15 pts). Let V, U and W be finite-dimensional inner product spaces
over F. Let S: U → V and T: V → W be linear transformations such that
im S = ker T.
Show that SS* + T*T is an isomorphism from V to V.