课程名称︰工程数学 - 线性代数
课程性质︰电机系必选
课程教师︰林茂昭
开课学院：电机资讯学院
开课系所︰电机工程学系
考试日期（年月日）︰2018/10/31
考试时限（分钟）：0910-1010
试题 :
Quiz 1 of Linear Algebra
1. (10%) Let u and v be any vectors in R^n.
Prove that the spans of {u, v} and {u + v, u - v} are equal.
2. (10%) Prove tha if A is an m×n matrix and B is an n×p matrix,
then rank AB ≦ rank B.
3. (10%) Prove that if A is an m×n matrix and P is an invertible m×m matrix,
then rank PA = rank A.
4. (10%) Let
T([x_1 x_2]^T) = [2x_1 + 3x_2, 4x_1 + 5x_2]^T. 其中 ^T 表转置矩阵
Determine whether T is one-to-one.
5. (10%) Let
T([x_1 x_2 x_3]^T) = [x_2 - 2x_3, x_1 - x_3, -x_1 + 2x_2 - 3x_3]^T
Determine whether T is onto.
6. (15%) Let A be an m×m matrix for which the i-th column is identical to the
j-th column, where i≠j. Prove that det(A) = 0
7. Let T: R^n → R^m ba a linear transformation and V ba subspace of R^n.
(a)(10%) Prove that W = {T(u): u \in V} is an subspace of R^m.
(b)(10%) Let {u_1, u_2, ..., u_k} be a basis of V.
Prove that {T(u_1), T(u_2), ...,T(u_k)} is a basis of W if T is
one-to-one.
8. (15%) Let \hat{B} = {b_1, b_2, b_3} ba a basis of R^3.
What is the relation between the matrix B = [b_1 b_2 b_3] and the
matrix A = [[e_1]_\hat{B} [e_2]_\hat{B} [e_3]_\hat{B}].