Is there exist some matrices $A,B$ that they have the same column space but different rank? (I do not get if this matrices is $n\times n$ or $m\times n$).

I think that they share the same column space something like this

A=$\begin{bmatrix} 1& 0\\ 0& 0 \end{bmatrix}$ B=$\begin{bmatrix} 1& 0\\ 0& 1 \end{bmatrix}$.

Here they have the same one vector from column space, but I do not know is this meaning that they need to have every vector of column space the same or not, what do you mean?

  • $\begingroup$ In your example, $\boldsymbol A, \boldsymbol B$ has different column spaces. $\endgroup$ – xbh Sep 2 '18 at 10:31

The rank of a matrix is equal to the dimension of the column space (and also equal to the dimension of the row space). Therefore, the answer is negative.

  • $\begingroup$ thank you for your help $\endgroup$ – Marko Škorić Sep 2 '18 at 10:32
  • $\begingroup$ @MarkoŠkorić If my answer was useful, perhaps that you could mark it as the accepted one. $\endgroup$ – José Carlos Santos Sep 2 '18 at 11:04
  • $\begingroup$ I try but I need to wait for 20 minutes, sorry $\endgroup$ – Marko Škorić Sep 2 '18 at 11:06

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