[Math] How does the span of vectors [1, 2] and [0,3] equal R2

linear algebravector-spacesvectors

I'm watching the video tutorial on spans here: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span

At 8:13, he says that the vectors a = [1,2] and b = [0,3] span R2. Visually, I can see it. But I tried to work it out, like so:

sp(a, b) = x[1,2] + y[0,3] such that x,y exist in R
         = [x, 2x] + [0, 3y] st x,y e R
         = [x, 2x + 3y] st x,y e R

With that said, how do we know that [x, 2x + 3y]spans R2? I tried picking a random point ([19, 6]) and let x=19 and solved for y (2*19 + 3y = 6) and found that when x=19 and y=-10, then I can get the point [19, 6].

But I'm confused as to how would I be able to find out that [x, 2x + 3y] can make any and all points in R2? What's my next step in determining if it spans R2?

Best Answer

Well then, let us pick any point in $\mathbb R^2$, e.g. the point $(a,b)$.

Let us define $x = a$ and $y=\frac{b-2a}{3}$ then you can easily see that:

$$ \begin{align*} (a,[b-2a]/3) = (x,y) \\ \iff (a,b-2a) = (x,3y) \\ \iff (a,b) = (x,2a+3y) \\ \iff (a,b) = (x,2x+3y) \end{align*} $$

So $(a,b)$ really is in the span of $\{(1,2),(0,3)\}$.

As we can choose any $(a,b) \in \mathbb R^2$, we know that those vectors span the whole $\mathbb R^2$.

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