1. Let us assume that matrix $B$ was created by deleting a number of rows and columns of matrix $A$. Does the rank of matrix $B$ can be larger than the rank of matrix $A$?

No, because some of them (rows and columns) could be dependent on each other, so deleting them may not decrease the rank of matrix.

  1. Are there such subspaces $V_1$ and $V_2$ of $\mathbb{R}^7$ that $dim(V_1 \cap V_2) = 2$ and $dim V_1 = dim V_2 = 5$ ?

No, there are not. For example: $V_1 = lin(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5)$, $V_2 = lin(\alpha_3, \alpha_4, \alpha_5, \alpha_6, \alpha_7)$ For those $2$ $dim(V_1 \cap V_2) = 3$ and they allready "use up" all possible dimensions.

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    $\begingroup$ For 2. your argument is not clear. For an argument see this post. $\endgroup$ – Dietrich Burde Nov 30 '20 at 13:51
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    $\begingroup$ Your argument for the first part isn't good either. Arguing that it is possible that the rank not go down does not prove that the rank can not go up. $\endgroup$ – lulu Nov 30 '20 at 13:58
  • $\begingroup$ Ok, you are right. But it actually can't (which was not written in my answer), am I right? $\endgroup$ – theboyboy Nov 30 '20 at 14:03

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