线代24春期中试题 en 发布版
1.(15 points, 3 points each) Multiple Choice. Only one choice is correct.
If are linearly dependent, then equals
(2) Iet be an real matrix and be an real column vector. Which of the following statements is correct?
(A) If does not have any solution, then has only the zero solution.
(B) If has infinitely many solutions, then has infinitely many solutions.
(C) If , both and have infinitely many solutions.
(D) If the rank of is , then has only the zero solution.
(3) For which value of does the system
(2) Let be a real matrix with rank 2 and Then the rank is
(4) Consider the system of linear equations:
The least-squares solution for the system is
- (10points)Let
Find an factorization of
- ( 24 points) Consider the following matrix and 4-dimensional column vector b:
(a) Find a basis for each of the four fundamental subspaces of (b) Find the complete solution to .
- (20 points) Let and be the linear transformation from defined by
Where denotes the vector space consisting of all real matrices. (a) Find the matrix representation of with respect to the following ordered basis
Find a matrix such that
(c) Find a matrix such that
- (5 points ) Let be two real matrices satisfying and Show that if , then Where denotes the zero matrix.
- (6 points) Let be a matrix, be a matrix such that
Find