课程名称︰应用数学一 课程性质︰系必修 课程教师︰陈义裕 开课学院：理学院 开课系所︰物理学系 考试日期（年月日）︰2007/04/17 考试时限（分钟）：180 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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