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）︰2008/04/15 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. (20 points) Please use Gaussian elimination to find the solution(s) to x + y - z + 3w + u = -2 2x + 3y + z + w - 2u = 8 -3x - y + 5z + w - u = 8 4x + 2y - 3z - 5w - 6u = -2 x - y + z + 2w + 3u = 3 Please note that you won't get any partial credits if you do not obtain the correct answer. 2. (30 points) Please answer if the following defines a vector space. If your answer is yes, then please also write down its dimension. Whenever your answer is no, please explicitly explain why not. (a) (6 points) V1 ≡ { x(5) | x(t) satisfies (d^2x / dt^2) = -x } (b) (6 points) V2 ≡ { (x,y,z,u) | x + y - z + 2w = 1 } t (c) (6 points) V3 ≡ { x(t) | dx/xt = -∫x(ξ)dξ, x(t = 1) = 0 } ︿ 0 ︿ (d) (6 points) V4 ≡ { L | all linear transformation L : R^3 → R^3 ︿ → → satisfying L(ey) = -ex } ︿ ︿ (e) (6 points) V5 ≡ { L | all linear transformation L : R^3 → R^3 ︿ ︿ L^2 = L } 3. (25 points) It is known that all the smooth functions f(x) defined on the interval x belongs [0, 2] form a vector space U. (a) (5 points) Let f1, f2, f3 belong U. Suppose f1, f2, f3 are linear independent. are f1 - 2*f2, 2*f2 - 3*f3, 3*f3 - f1 linearly independent? Please prove your claim. ︿ (b) (5 points) Is the mapping F : U → U defined by ︿ F( f(x) ) ≡ (x^2 * sin(πx)) * f(x) a linear transformation? Please prove your claim. (c) (5 points) Please prove that U is not a finite-dimensional vector space. (d) (10 points) Please show that 1, x and x^2 are linearly independent vectors in U. 4. (25 points) It is known that 1, x amd x^2 span a three-dimensional space U (so that U consists of all the polynomials of degree less than 3). Use them as a basis for U. ︿ (a) (10 points) Pleaase write down the matrix representation of L : U → U defined by 　　　　　　　　　　　 ︿　 L(f(x)) ≡ ( df(x) / dx ) + f(x). (b) (5 points) If we decide to use 1 + x^2, x, and 1 - x^2 as the new basis. Please find the matrix representation of L in this new basis. (c) (10 points) One can show that L is invertible. Please find ︿-1 L(x^2). --

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