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…
Best Answer
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.