课程名称︰线性代数 课程性质︰系必修 课程教师︰陈文进 开课学院：电资学院 开课系所︰资工系 考试日期（年月日）︰2010/1/14 考试时限（分钟）：180min 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.(8%)Let a,b,u,v,w€R^4. A = [a u v w], and B = [b u v w] be two 4x4 matrices , where det(A) = 4 and det(B) = 1. Find det(A+B) ┌0 0 0┐ 2.(10%)Let A, B, and P be 3x3 matrices, where B = │1 0 3│ and AP = PB. └0 1-2┘ Find det(A+I3) 3.(12%)Let V be the plane defined by 2x+y-3z=0. (a) Find an orthogonal basis for V. (b) Use (a) to find the distance of the point(2,2,1) to V. 4.(a)(6%)Find the least squares solution of { x1+x2=4 { 2x1+x2=-2 { x1-x2=1 (b)(6%)Use (a) to find the orthogonal projection of the point (4,-2,1) to the plane spanned by vectors [1,2,1]^T and [1,1,-1]^T € R^3 5.(10%)Let P[x]={a+bx|a,b€R} be the vector space consists of polynomials of degree 1. The inner product of the two functions f(x),g(x)€P[x] is defined as <f,g> = ∫(0 to 1) f(t)g(t)dt. Find the orthogonal projection of h(x)=4+3x-x^2 on P[x]. 6.(10%)Let A and B be two nxn matrices. Prove that AB and BA have the same set of eigenvalues. 7.(10%)If a[0]=2, a[1]=3, and a[k+1]=3a[k]-2a[k-1], for all k>=1. Use method of linear algebra to find the formula of a[k]. 8.(a)(8%)Find the standard form of the quadratic curve 3(x^2)+2xy+3(y^2)+6√2 x-10√2 y+10=0. (b)(4%)Give the coordinates of the center and the vectors representing the two principal axes of the curve.(the x'-axis must be 0<=θ<π/2 and the y'-axis must be π/2<=θ<π) 9.(a)(8%)Find the SVD of A = ┌3 2 2┐ └2 3 -2┘ (b)(8%)Use (a) to find an orthogonal basis for each of the four subspaces R(A), C(a), N(A), and N(A^T). -- 心中有变数，就不用变数代换法 --

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