linear algebra proofs

**deniselim17** · April 2nd 2010, 03:59 AM

I want to show that $rank(A+B) \leq rank(A)+rank(B)$ where $A,B \in M_{m,n}(R)$ .

It is obvious that $rank(A), rank(B), rank(A+B)$ are $\leq m$ . So, $rank(A)+rank(B) \leq 2m, rank(A+B) \leq m$ . Can I conclude now that $rank(A+B) \leq rank(A)+rank(B)$ ? I think there should be somemore steps to continue this proof.

Next questions, I need some hints.
Let $u_{1},...,u_{k},v_{1},...v_{n} \in R^{n}$ . If each vector $u_{i}$ is a linear combination of vectors of $v_{1},...,v_{n}$ . Show that the maximum number of linearly independent vectors in $\{u_{1},...,u_{k}\}$ cannot exceed the maximum number of linearly independent vectors in $\{v_{1},...,v_{n}\}$

**deniselim17** · April 2nd 2010, 06:47 PM

it's urgent. Can anyone help???

**deniselim17** · April 2nd 2010, 09:25 PM

I proved both of them.
Thank you anyway.

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