How do I approach linear algebra proving problems in general? I have massive problems with questions like these:
Let $\{v_1, . . . , v_r\}$ be a set of linearly independent vectors in $\mathbb{R}^n$
(with
$r < n$), and let $w\in\mathbb{R}^n$  be a vector such that $w \in \mathrm{span}\{v_1, . . . , v_r\}$.
Prove that $\{v_1, . . . , v_r, w\}$ is a linearly independent set.
Let $U$ and $V$ be subspaces of $\mathbb{R}^n$
Define the set $U + V = \{u + v|u ∈ U, v ∈ V \}$. Prove that $U + V$
is a subspace of $\mathbb{R}^n$.
I'm not looking for the answers to these 2 questions but instead I want to know how do I learn to approach these problems. These proving problems are my absolute Achilles' heel. I can't get the solution at all. What can I do to learn to solve these problems? Any online resources you guys can recommend? I usually learn how to do real questions by following examples but I get nothing from watching people talk about these theories and principles behind how it's done...
 A: Linear Algebra questions are usually based on the definition of linear independence. Once you get the hang of it, the problems make much more sense.
For a good on-line resource, you could try Khan Academy.
A: In general, there is no sure work formula to  successfully prove something and there could be multiple ways to solve the same thing . The advices in the comments are very useful. 
Reading mathematics does help if you can see the thought process.
Mathematical proofs is really like solving real life problem.
I usually ask myself questions to guide myself to solve problems:


*

*what is my goal? 

*what tools do I have? 

*how do i use my tools to reach my goal?

*if I get stucked, can I use tricks like contradiction or contrapositive to prove it. Do I have other lemmas or theorems that can help me achieve my goal.

*Sometimes, you really get stucked, ask for help/ hint from tutor/ friends. Take a walk around and come back to the question later. Try to solve special cases or look at similar questions.


For example, in the first question:
Let $\{v_1, . . . , v_r\}$ be a set of linearly independent vectors in $\mathbb{R}^n$
(with
$r < n$), and let $w\in\mathbb{R}^n$  be a vector such that $w \notin \mathrm{span}\{v_1, . . . , v_r\}$.
Prove that $\{v_1, . . . , v_r, w\}$ is a linearly independent set.
Goal: prove that $\{v_1, . . . , v_r, w\}$ is a linearly independent set.
hmmm... what does linearly independent set mean?  Let me check the definition and rewrite it.
New goal: If $\sum_{i=1}^r c_iv_i+c_{r+1}w = 0$, show that $c_i=0, \forall i=1, \ldots, r+1.$
Now let's see what do we know about $v_i$ and $w$.
Tool $1$: Let $\{v_1, . . . , v_r\}$ be a set of linearly independent vectors in $\mathbb{R}^n$
(with
$r < n$). Meaning, whenever I form a linear combination of $v_i$ and equate it to $0$, the coefficients must be $0$.
Tool $2$: $w \notin \mathrm{span}\{v_1, . . . , v_r\}$, that is $w$ cannot be written as linear combination of $v_i$.
The second tool motivates me to try to isolate $w$.Hence, it is natural to consider whether :


*

*case 1: $c_{r+1}=0$: If $\sum_{i=1}^r c_iv_i+c_{r+1}w = 0$ and $c_{r+1}=0$, we have $\sum_{i=1}^r c_iv_i = 0$ and we can check our tool list and see that we can use tool $1$ to conclude that $c_i=0, \forall i=1,\ldots, r$.

*case 2: $c_{r+1}\neq0$: We better get a contradiction for this case, as we really want $c_{r+1}$ to be zero. Suppose it is not, $w =-\frac{1}{c_{r+1}}\sum_{i=1}^r c_iv_i$, check our tool list and we can see that tool $2$ says that we can't have this case.

