课程名称︰线性代数 课程性质︰必修 课程教师︰(统一教学) 开课学院：电资学院 开课系所︰电机系 考试日期（年月日）︰2009.04.22 考试时限（分钟）：100 分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1.(28%) Label the following statements as being true or false. (Explain your answer. Answers with no explanation get 0%): (a) Let A be an m*n matrix with reduced row echelon form R. Then there is a unique matrix P such that PA = R. (b) If T is a linear transformation, then the dimension of the range T plus the dimension of the null space of T equals the dimension of the domain of T. (c) If A is an n*n matrix and the system Ax = b is consistent for every b, then A is invertible. (d) Let S be a linearly independent subset of R^n and V be a k-dimensional subspace of R^n. If S has k vectors, then S is a basis of V. (e) The vectors in the vector form of the general solution to Ax = 0 form a basis for the null space of A. (f) Let V and W be two subspaces of R^2. V∪W is a subspace of R^2. (g) The pivot columns of the reduced row echelon form of A form a basis for the column space of A. 2.(15%) Find the determinant of the n*n matrix ┌ ┐ │ a+b ab 0 0 … 0 0 0 │ │ 1 a+b ab 0 … 0 0 0 │ │ 0 1 a+b ab … 0 0 0 │ A = │ … … … … … … … … │ │ 0 0 0 0 … 1 a+b ab │ │ 0 0 0 0 … 0 1 a+b │ └ ┘ 3.(15%) Find an explicit description of the reflection T of R^2 about the line with equation y = mx. 4. Let T and U be linear transformations with standard matrices A and (A^T)A, respectively. Let T be onto. (a)(7%) Show whether U is onto, on-to-one, or invertible. (b)(7%) What is the dimension of the null space of U? Explain your answer. 5.(8%) Prove that for any m*n matrix A and any n*p matrix B, rank AB ≦ rank A. 6. Let T be the linear opeartor on R^3 such that ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ │1│ │1│ │ 0│ │ 2│ │-1│ │-1│ T(│0│) = │1│, T(│-1│) =│-1│, T(│ 1│) =│ 0│ │1│ │3│ │ 1│ │ 0│ │-1│ │-1│ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘ └ ┘ ┌ ┐ ┌ ┐ ┌ ┐ │1│ │1│ │3│ B = { │1│,│1│,│2│} is a basis for R^3. │0│ │1│ │1│ └ ┘ └ ┘ └ ┘ (a)(10%) Find the B-matrix representation of T. (b)(10%) Find a basis for the null space of T and represent it in B-coordinate vectors. --

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