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(a)
A matrix with column space contains
The required matrix is
Given:
Column space is
Calculation:
Calculate the column numbers and row numbers individually,
Therefore, the required matrix formed is
(b)
A matrix with column space has basis
The required matrix would be impossible to create because the value of addition of dimensions is not 3.
Given:
Column space has basis
Calculation:
To construct a matrix, the dimension of the column space and the dimension of the nullspace must sum up to 3, which is not possible with the column space has basis
Therefore, the required matrix is not possible.
(c)
A matrix with dimension of nullspace
The required matrix is
Given:
Dimension of nullspace
Calculation:
The dimension of nullspace is one more than the dimension of left nullspace.
Therefore, the required matrix could be formed as
(d)
A matrix with nullspace contains
The required matrix is
Given:
Nullspace contains
Calculation:
The left nullspace contains
Therefore, the matrix formed is the required matrix is
(e)
A matrix when Row space
The required matrix is not possible.
Given:
Row space
Calculation:
Row space
Consider
However,
Therefore, the required matrix is not possible.
(b)
A matrix with column space has basis
The required matrix would be impossible to create because the value of addition of dimensions is not 3.
Given:
Column space has basis
Calculation:
To construct a matrix, the dimension of the column space and the dimension of the nullspace must sum up to 3, which is not possible with the column space has basis
Therefore, the required matrix is not possible.
(c)
A matrix with dimension of nullspace
The required matrix is
Given:
Dimension of nullspace
Calculation:
The dimension of nullspace is one more than the dimension of left nullspace.
Therefore, the required matrix could be formed as
(d)
A matrix with nullspace contains
The required matrix is
Given:
Nullspace contains
Calculation:
The left nullspace contains
Therefore, the matrix formed is the required matrix is
(e)
A matrix when Row space
The required matrix is not possible.
Given:
Row space
Calculation:
Row space
Consider
However,
Therefore, the required matrix is not possible.
(c)
A matrix with dimension of nullspace
The required matrix is
Given:
Dimension of nullspace
Calculation:
The dimension of nullspace is one more than the dimension of left nullspace.
Therefore, the required matrix could be formed as
(d)
A matrix with nullspace contains
The required matrix is
Given:
Nullspace contains
Calculation:
The left nullspace contains
Therefore, the matrix formed is the required matrix is
(e)
A matrix when Row space
The required matrix is not possible.
Given:
Row space
Calculation:
Row space
Consider
However,
Therefore, the required matrix is not possible.
(d)
A matrix with nullspace contains
The required matrix is
Given:
Nullspace contains
Calculation:
The left nullspace contains
Therefore, the matrix formed is the required matrix is
(e)
A matrix when Row space
The required matrix is not possible.
Given:
Row space
Calculation:
Row space
Consider
However,
Therefore, the required matrix is not possible.
(e)
A matrix when Row space
The required matrix is not possible.
Given:
Row space
Calculation:
Row space
Consider
However,
Therefore, the required matrix is not possible.
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Chapter 3 Solutions
Introduction to Linear Algebra, Fifth Edition
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