# 2 questions on linear algebra. Would like to evaluate my answers.

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.

• For 2. your argument is not clear. For an argument see this post. – Dietrich Burde Nov 30 '20 at 13:51
• 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. – lulu Nov 30 '20 at 13:58
• Ok, you are right. But it actually can't (which was not written in my answer), am I right? – theboyboy Nov 30 '20 at 14:03