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.

Please check you haven't yet proved anything... . You could argue as follows: Suppose $\displaystyle 0=a_1u_1+a_3u_3=a_1u_1+0\cdot u_2+a_3u_3\Longrightarrow a_1=0=a_3$ since we're given $\displaystyle \{u_1,u_2,u_3\}$ is lin. ind.
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.

This is similar as the above one but at least here you mentioned that the given set of vectors is l.i....I'd wrap it up as follows: Since $\displaystyle \{u_{1}, u_{2},u_{3}\}$ is l.d. we have a trivial linear combination $\displaystyle a_1u_1+a_2u_2+a_3u_3=0$ and $\displaystyle a_i\neq 0$ for some $\displaystyle 1\leq i\leq 3\Longrightarrow 0\cdot u+a_1u_1+a_2u_2+a_3u_3=0$ and STILL $\displaystyle a_i\neq 0$ for some index $\displaystyle i\,\Longrightarrow \{u,u_1,u_2,u_3\}$ is l.d.
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?