課程名稱︰應用數學一 課程性質︰系必修 課程教師︰陳義裕 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2007/04/17 考試時限（分鐘）：180 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. (15 points) Please use Gaussian elimination to find the solution(s) to x + y - z + 3w = 5 2x + 3y + z + w = 3 -3x - y + 5z + w = -1 4x + 2y - 3z - 5w = -5 Please note that you won't get any partial credits if you do not obtain the correct answer. 2. (20 points) Please answer if the following defines a vector space. There is no need for you to prove or explain it if you think it is a vector space, but you will have to explicitly explain why not if your answer is no. (a) (5 points) V1 ≡ { x(t) | (d^2/dt^2)x = - x - x^3, x(t = 0) = 0, (dx/dt)| = 0 } t=0 (b) (5 points) V2 ≡ { (x, y, z, u) | x + y - z + 2w = 1 } (c) (5 points) V3 ≡ { x(t) | (d^2/dt^2)x = -x, x(t = π) + (dx/dt)| = 0} t=0 ┌ ┐ │x 1 y│ (d) (5 points) V4 ≡ { all matrices of the form │z w 0│, where └ ┘ x + y - z + w = 0 } 3. (30 points) All the smooth functions f(x) defined on the interval x belongs to [0, 2] form a vector space U. (a) (10 points) Please show that 1, x and x^2 are linearly independent vectors in U. (b) (10 points) Are sin(πX) and sin(π(x - 1)) linearly independent vector in U? Either way, you will have to prove it. (c) (5 points) Is the mapping F : U → U defined by F(f(x)) ≡ sin(πx) * f(x) a linear transformation? Please prove your claim. ︿ (d) (5 points) It is known that L defined below is a linear transformation: ︿ L(f(x)) = ∫f(x)dx. ︿ What is the dimension of the image of L? Please prove your claim. → → → 4. (20 points) Let {e1, e2, e3} be a basis of a vector space U and let → → {ε1, ε2} be a basis of a vector space V. It is known that a linear ︿ transformation L : U → V has the following matrix representation. ┌ ┐ │1 3 -2│ │2 4 3│ └ ┘ → → (a) (10 points) Someone decides to change the basis of V from {ε1, ε2} → → to {ε1', ε2'}, where → → → ε1' ≡ ε1 + ε2 → → → ε2' ≡ ε1 - 2*ε2 ︿ Please explicitly compute the matrix representation of L in this new basis. → → → (b) (10 points) Someone else instead decides to use {e1', e2', e3'} as a → → new basis of U but keep the same old {ε1, ε2} as the basis of V, where → → → e1' ≡ e1 = e2 → → → e2' ≡ e1 - 2*e2 → → e3' ≡ e3 ︿ Please explicitly compute the matrix representation of L in this new basis. 5. (15 points) According to Part (a) of Problem 3, we know that 1, x and x^2 span a three-dimensional subspace in U. (Even if you did not answer that part correctly, you can still work on the present problem. They are separated!) Call this subspace V. Now consider the linear transformation ︿ L : V → V satisfying ︿ L(1) = x^2 + x + 1 ︿ L(x) = x - 1 ︿ L(x^2) = x^2 + 2x (a) (5 points) Please write down the matrix representation of L if we use 1, x and x^2 as the basis of V. ︿ (b) (10 points) Please find the kernel of L, i.e., all the functions f(x) 　　　　　　　　　　　　 ︿ in V that satisfy L(f(x)) = 0 --

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