If I find a basis for the column space of A, will that basis also be a basis for the column space of col(A transpose) ?
Basis of column space equivalence to transpose row space?
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linear-algebra
change-of-basis
1 Answers
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This is not true in general. First, if $A$ is not a square matrix then the column space and the row space live in vector spaces of different dimension. But even if it is a square matrix, then this claim is not true. For example $$\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right)$$ has column space spaned by $\left(\begin{array}{c} 1\\ 0 \end{array}\right)$ while $A^T$ has column space spanned by $\left(\begin{array}{c} 0\\ 1 \end{array}\right)$. While these spaces are not the same, they do have the same dimension, and this is always true.