How to find whether $(2,1,3)$ and $(-1,3,6)$ belong to subspace spanned by vectors $x=(1,2,3)$ and $y=(-1,3,6)$

Question

How to find whether $(2,1,3)$ and $(-1,3,6)$ belong to subspace spanned by vectors $x=(1,2,3)$ and $y=(-1,3,6)$.

I'm new to vector space and struggling with it to find its solution.

Answer 1

We already know that $(1,2,3)$ and $(-1,3,6)$ are linearly independent (Why?)

Because a collection of two vectors are dependent on one another if and only if one is a scalar multiple of the other.

Now... $(-1,3,6)$ (from the beginning of your sentence) is obviously in the subspace spanned by $x=(1,2,3)$ and $y=(-1,3,6)$ (from the end of your sentence) because it is a linear combination of the basis vectors... specifically equal to $0x+1y$. No calculation was needed to see this since it was equal to one of the basis vectors it was clearly in the span.

As to the question of if $(2,1,3)$ is in the span, there are many approaches. We could try to find values of $a$ and $b$ such that $(2,1,3)=ax+by$. We could try row reducing $\begin{bmatrix}1&2&3\\-1&3&6\\2&1&3\end{bmatrix}$ or the transpose of that and counting the number of pivots. We could try finding the determinant of that matrix as well. There are several other approaches possible too. Pick whichever you like best.

(Row reducing might be the most general approach as it would work in the event that the matrix produced wasn't square)