线代23秋期中试题 en 发布版
1.(15 points, 3 points each) Multiple Choice. Only one choice is correct.
(1)Suppose that are a basis for nullspace of a matrix . Which of the following lists of vectors is also a basis for ?
(A).
(B),
(C).
(D).
(2)Which of the following statements is correct?
(A) If the columns of are linearly independent, then has exactly one solution for every b.
(B) Any matrix has linearly dependent columns.
(C) If the columns of a matrix are linearly dependent, so are the rows.
(D) The column space and row space of a matrix may have different dimensions.
(3)Let
can be written as a linear combination of if
(A)1.
(B)3.
(C)6.
(D)9.
(4) Which of the following statements is correct?
(A) Suppose that and is an invertible matrix, then the column space of and the column space of are the same.
(B) Let be a matrix with rank 1, then , where is a positive integer and is a real number.
(C) Let be symmetric matrices, then is symmetric.
(D) If is of full row rank, then has only the zero solution.
(5)Let A and B be two matrices. If A is a non-zero matrix and AB = 0, then
(1)
(2)
(3)
(4).
2.
(1)Denote the vector space of real matrices by , and let be the subspace of consisting of skew-symmetric real matrices, then dim
(2) be two invertible matrices. Suppose the inverse of is，where is the zero matrix. Then
(3)Let with . Then
(4)Consider the system of linear equations
The least-squares solution for the system is
(5) Let be the subspace of be defined as follows:
A unit vector orthogonal to is
3.(24 points) Consider the following matrix A with its reduced row echelon form R:
(a) Find a basis for each of the four fundamental subspaces of (b) Find the third column of matrix
4.(15points)Let
For which value(s) of , the matrix equation has no solution, a unique solution, or infinitely many solutions? Where is a matrix. Solve if it has at least one solution.
5.(15 points) Let be the vector space of real matrices. Let
Consider the map
for any real matrix , where denotes the trace of a matrix The trace of an matrix D is defined to be the sum of all the diagonal entries of D, in other words,
(a) Show that is a linear transformation.
(b) Find the matrix representation of with respect to the ordered basis
and the standard basis
for
(c) Can we find a matrix such that If yes, please find one such matrix. Otherwise, give an explanation.
6.(6 points) Let be an matrix, be an matrix, and be an matrix. Show that
where is the zero matrix.