FALSE.
Counterexample:
A = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] {\displaystyle A={\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{bmatrix}}}
For this matrix, we have dim ( R ( A ) ) = 5 {\displaystyle \displaystyle {\text{dim}}(R(A))=5} since there are 5 pivot columns.
By the rank-nullity theorem,
dim ( N ( A ) ) = 5 − dim ( R ( A ) ) = 5 − 5 = 0 {\displaystyle \displaystyle {\text{dim}}(N(A))=5-{\text{dim}}(R(A))=5-5=0}
Thus dim N(A) ≥ 3 {\displaystyle {\text{dim N(A)}}\geq 3} is not true.
since there are 5 pivot columns.
is not true.
