Assuming we have matrix B as follows: $$ B=\begin{bmatrix} b_{11}&\cdots&b_{1n}\\ b_{21}&\cdots&b_{2n}\\ \vdots&\vdots&\vdots\\ b_{n1}&\cdots&b_{nn} \end{bmatrix}$$ I want to verify the space spanned by columns of B ,n$\times$n matrix, is equivalent to the space spanned by column of A ,a m$\times$ m matrix as follows: $$A= \begin{bmatrix} b_{11}&\cdots&b_{1n}&0&0...&0\\ b_{21}&\cdots&b_{2n}&0&0...&0\\ \vdots&\vdots&\vdots&0&0...&0\\ b_{n1}&\cdots&b_{nn}&0&0...&0\\ 0&\cdots&0&1&0&...&0\\ 0&\cdots&0&0&1&...&0\\ \vdots&\vdots&...&&&\vdots\\ 0&\cdots&0&0&0&...&1\\ \end{bmatrix}$$ I cannot see any reasons to contradict my assumption(Since the the dimension of column vectors of A and B are not the same I was not sure).
Could anyone help please?
Thanks in advance.