determine whether a set is a vector space Let $V$ be the set of all $\mathbb{R}^4$ vectors of the form
$$\begin{bmatrix}
 a − 2b + 5c \\
 2a + 5b − 8c\\
−a − 4b + 7c\\
 3a + b + c
\end{bmatrix}$$
for $a, b, c \in \mathbb{R}$. Explain
why $V$ is a vector space and find a basis for $V$
In this case, how am I supposed to determine whether the set is a vector space or not? Do I need to go through the ten axioms and check if it does hold all of the rules? Is there other simpler ways to approach this one?
 A: You don't have to go through all the ten axioms. You can show that it's a vector sub-space of $\mathbb{R}^4$ by just showing that it's closed under $\lambda_1 \vec{v} + \lambda_2 \vec{w}$ for any $\vec{v},\vec{w} \in V$ and $\lambda_1,\lambda_2\in \mathbb{R}$. That proves that $V$ is a vector space itself.
Finding a spanning set is easy. Notice that:
$$(a − 2b + 5c,
2a + 5b − 8c,
−a − 4b + 7c,
3a + b + c) = a(1,2,-1,3)+b(-2,5,-4,1)+c(5,-8,7,1)$$
So, every vector in $V$ is a linear combination of the three vectors $(1,2,-1,3),(-2,5,-4,1),(5,-8,7,1)$.
Show that these three vectors are linearly independent and therefore, the following set is a basis:
$$\{(1,2,-1,3),(-2,5,-4,1),(5,-8,7,1)\}$$
You can do this by doing elementary row operations on the $3\times 4$ matrix that is obtained by putting each of these vectors in each row and reducing it to the form $[I_3,b]$ by elementary operations only. If you can do this, you have proved that they're linearly independent and therefore, they form a basis because they span the set and are linearly independent. 
