课程名称︰线性代数 课程性质︰大一必修 课程教师︰ 开课学院： 开课系所︰电机系 考试日期（年月日）︰96/04/16 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. (21%) Label the following statememts as being true or false. (Explain your answer. Answers with no explanation get 0%): (a) Let A be an m × n matrix with reduce row echelon form R. If rank A = N then there is a unique matrix P such that PA = R. (b) If (A^T)A = O, then A = O. (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) If a linear transformation is one-to-one, then it is invertible. (f) Let V and W be two subspace of R^2. V∪U is a subspace of R^2. (g) Let R be the reduced row echelon form of A. Then R and A have the same column space. 2. (9%) Find a basis for the column space and the null space of the following matrix. [ -1 2 1 -1 -2 ] [ 2 -4 1 5 7 ] [ 2 -4 -3 1 3 ] 3. Given the matrix a below, find the determinants of (a) (8%) A, (b) (2%) A^(-1), (c) (3%) 2A, and (d) (2%) R where R is the reduced row echelon form of A. [ 0 -1 2 -1 ] A = [ 1 0 -1 1 ] [ 1 -1 1 -1 ] [ -1 1 1 0 ] 4. (10%) Let V and W be two subspaces of R^n. Show whether V + W = {sεR^n : s = v+ w, vεV and wεW} is a subspace of R^n. 5. Let T: R^3 → R^3 be the linear transformation defined by ( [ x1 ] ) [ x1 ﹣x2 + x3 ] T ( [ x2 ] ) = [ -x1 + x2 ﹣3x3 ] ( [ x3 ] ) [ 2x1 + x3 ] (a) (2%) Find the standard matrix of T. (b) (4%) Is T one-to-one? Is T onto? (c) (9%) Is T invertible? If it is, find T^-1. 6. (a) (5%) Let A be a 3×3 matrix such thea A^-1 = -A. Show that whether A is invertible? (b) (5%) Let C be a 3×3 matrix with det C = 2. What is the determinant of B = [ C -C ]? [ C C ] 7. Let V1 be a 3-dimensional subspace of R^4 and V2 be a 2-dimensional subspace of R^4. Let W be the intersrction of V1 and V2, i.e., W = V1∩V2. (a) (6%) Prove that W is also asubspaceof R^4. (b) (8%) Prove that W contains some nonzero vectors. (c) (6%) Let V1 and V2 be respectively the span of S1 and S2 given below. Find a basis for W. { [ 1 ] [ -1 ] [ 2 ] } { [ 1 ] [ 2 ] } S1 = { [ -1 ],[ 1 ] [ 2 ] }, S2 = { [ 3 ],[ 6 ] } { [ 2 ] [ -3 ] [ 2 ] } { [ 0 ] [ 3 ] } { [ -3 ] [ 2 ] [ 0 ] } { [ 3 ] [ 9 ] } --

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