(a)
The same subspaces for matrices of different sizes and prove that all matrices have same rank r.
Yes, the row space and nullspace are same and rank r is also same.
Given:
Calculation:
Consider the matrix
It is observed that the row space and the nullspace both are same, so this means that the rank r, i.e. dimension of row space, is also same.
Therefore, the row space and nullspace are same and rank r is also same.
(b)
The same subspaces for matrices of different sizes and prove that all matrices have same rank r.
Yes, the column space and nullspace are same and rank r is also same.
Given:
Calculation:
Consider the matrix
It is observed that the column space and the nullspace both are same, so this means that the rank r, i.e. dimension of column space, is also same.
Therefore, the column space and nullspace are same and rank r is also same.
(b)
The same subspaces for matrices of different sizes and prove that all matrices have same rank r.
Yes, the column space and nullspace are same and rank r is also same.
Given:
Calculation:
Consider the matrix
It is observed that the column space and the nullspace both are same, so this means that the rank r, i.e. dimension of column space, is also same.
Therefore, the column space and nullspace are same and rank r is also same.
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Introduction to Linear Algebra, Fifth Edition
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