linearly independent and dependent

Problem 1:

If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly independent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u_{1}, u_{3} \}$ is linearly independent set.

Proof 1:

If $\displaystyle a_{1}u_{1}+a_{3}u_{3}=0$ where $\displaystyle a_{1}, a_{3} \in R$, $\displaystyle a_{1}=a_{3}=0$ since $\displaystyle u_{1},u_{2},u_{3}$ is linearly independent.

Is the proof correct? I felt something missing in the proof.

Problem 2:

If $\displaystyle u_{1}, u_{2},u_{3}$ are linearly dependent vectors in $\displaystyle R^{n}$, show that $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent set for any $\displaystyle u \in R^{n}$.

Proof 2:

If $\displaystyle au+a_{1}u_{1}+a_{2}u_{2}+a_{3}u_{3}=0$. Since $\displaystyle u_{1}, u_{2}, u_{3}$ are linearly dependent, $\displaystyle a_{i} \neq 0$ for some $\displaystyle i=1,2,3$. Thus, $\displaystyle \{u, u_{1}, u_{2}, u_{3} \}$ is linearly dependent.

Please check my proofs.

Problem 3:

Let $\displaystyle u_{1}, ..., u_{k} \in R^{n}$ and $\displaystyle A \in M_{n}(R)$, $\displaystyle A$ is invertible .

Show that $\displaystyle u_{1}, ..., u_{k}$ are linearly independent if and only if $\displaystyle u_{1}A, ..., u_{k}A$ are linearly independent vectors in $\displaystyle R^{n}$.

How to prove problem 3? what theorem do I need?