Subset, Dimension and Vector Space 
Suppose that $\{v_1,v_2,v_3\}$ is a linearly independent subset of a vector space $V$, with $\dim(V) = 4$, and that $v_4 \not\in \text{span}\{v_1,v_2,v_3\}$. Prove that $\{v_1,v_2,v_3,v_4\}$ is a basis for $V$.

My current answer would be:
Since $\dim(V) = 4$, it requires another vector (which is not linearly dependent) to form a basis.
Hence $v_4$ is not equal to $k_1 v_1+k_2 v_2+k_3 v_3$ (except if $k1=k2=k3 = 0$)
Since $v_4$ is not in $\text{span}\{v_1,v_2,v_3\}$, it is not linearly dependent and $\{v_1,v_2,v_3,v_4\}$ is a basis for $V$ (since vectors as a basis should be linearly independent).
Just wondering if you can help me whether my answer respond to the question well. Any help will be appreciated.
Thank you very much
 A: The idea of your proof is correct, but it should be stated more clearly.
First, since $v_4 \not\in \text{span}\{v_1,v_2,v_3\}$, this means that there are no scalars $k_1,k_2,k_3$ for which $v_4 = k_1 v_1 + k_2 v_2 + k_3 v_3$, and this is remains true if $k_1 = k_2 = k_3 = 0$. You don't need to treat that as a special case. It is only when discussing linear independence that we need to exclude that case.
Then, to argue that $\{v_1,v_2,v_3,v_4\}$ is linearly independent, suppose for a contradiction that this is false. This means that there are scalars $k_1,k_2,k_3,k_4$, not all zero, such that
$$k_1 v_1 + k_2 v_2 + k_3 v_3 + k_4 v_4 = 0$$
Now $k_4$ can't be zero, since otherwise this equation becomes
$$k_1 v_1 + k_2 v_2 + k_3 v_3 = 0$$
which is only true if $k_1 = k_2 = k_3 = 0$ since $\{v_1,v_2,v_3\}$ is a linearly independent set. Therefore, since $k_4 \neq 0$, we can divide by $k_4$ and rearrange to obtain
$$v_4 = -\frac{k_1}{k_4} v_1 - \frac{k_2}{k_4}v_2 - \frac{k_3}{k_4}v_3$$
But this expresses $v_4$ as a linear combination of $v_1,v_2,v_3$, so $v_4 \in \text{span}\{v_1,v_2,v_3\}$, which is a contradiction.
We conclude that $\{v_1,v_2,v_3,v_4\}$ must be linearly independent. Moreover, it is a linearly independent set of $4$ vectors, in a vector space of dimension $4$, hence it is a basis for $V$.
