(a)
The matrix that leads X to
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
Consider row reduction method to check that the rows of A are dependent and A has rank 2.
The nullspace of A has basis:
Thus, each column of X is in the nullspace of A, that means when they are multiples of the basis of
(b)
The matrix that have the form AX for some matrix X.
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the vectors whose components add to zero, that means,
(c)
The dimensions of nullspace and column space of M. Also, reason behind
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.
Thus, dimension of a 3 by 3 matrix M will be
Therefore,
(b)
The matrix that have the form AX for some matrix X.
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
The columns of any matrix form AX are linear combinations of the columns of matrix A. Thus, these are the
(c)
The dimensions of nullspace and column space of M. Also, reason behind
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.
Thus, dimension of a 3 by 3 matrix M will be
Therefore,
(c)
The dimensions of nullspace and column space of M. Also, reason behind
Given:
M is the space of 3 by 3 matrices and multiply every matrix X in M by
Calculation:
It is noted that dimension of the nullpace of M is 3 and the dimension of column space B is 6.
Thus, dimension of a 3 by 3 matrix M will be
Therefore,
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Introduction to Linear Algebra, Fifth Edition
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