How to check if a set of vectors is a basis

A set of vectors $v_1, v_2, ..., v_n$ is linearly independent if and only if we have that

$$a_1v_1 + a_2v_2 + ... + a_nv_n = 0 \;\;$$

only when $ a_1 = a_2 = ... = a_n = 0 $.

(After all, any linear combination of three vectors in $\mathbb R^3$, when each is multiplied by the scalar $0$, is going to be yield the zero vector!)

So you have, in fact, shown linear independence. And any set of three linearly independent vectors in $\mathbb R^3$ spans $\mathbb R^3$. Hence your set of vectors is indeed a basis for $\mathbb R^3$.

Your confusion stems from the fact that you showed that the homogeneous system had only the trivial solution (0,0,0), and indeed homogeneous systems will always have this solution. The criteria for linear dependence is that there exist other, nontrivial solutions.

Another way to check for linear independence is simply to stack the vectors into a square matrix and find its determinant - if it is 0, they are dependent, otherwise they are independent. This method saves a bit of work if you are so inclined.

The easiest way to check whether a given set $\{(a,b,c),(d,e,f),(p,q,r)\} $of three vectors are linearly independent in $\Bbb R^3$ is to find the determinant of the matrix, $$\begin{bmatrix} a & b & c\\ d & e& f\\ p&q&r \end{bmatrix}$$ is zero or not.

If the determinant is zero then the set is linearly dependent else i.e. determinant is nonzero it is linearly independent. Which is the same as saying that the system of linear equations, $$\begin{bmatrix}a&b&c\\d&e&f\\p&q&r\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$$ has only the trivial solution.