[Math] Describe the set span geometrically

linear algebravectors

So I have 2 questions that both ask to describe a set span geometrically…

  1. Show that $span((1,2))$ does not $= R^2$. Describe the set $span ((1,2)) $geometrically.

  2. If u $\in R^2$, describe the set span (U) geometrically. (Hint: Do not forget a very special case).

I believe I'm trying to describe the shape of the span(Is that what this question is asking?), but I don't understand how you do that. When trying to figure this out, I've read that "One vector spans a line. Two linearly independent vectors span a plane. And ≥3 linearly independent vectors span a hyper-plane." So do I use on of these ideas to describe it? Could some please explain how you describe a set span geometrically?

Also with question one, how do you prove a span is not in $R^2$? I understand how to prove something is in $R^2$, but not when it's not. Could some also please explain how you would do that?

Best Answer

  1. To do this, we just have to find an element of $\mathbb{R}^2$ that can't be written as a multiple of $(1,2)$, since the set of multiples of $(1,2)$ is precisely the set span$((1,2))$. One such element is $(2,2)$: there is no scalar $a$ such that $(2,2) = a(1,2)$. To give a geometric description, use what you said about one vector spanning a line. Think about what happens when you have one vector and take the set of all scalar multiples of this vector: what does it look like?

  2. Assuming that you meant to write "describe the set span$(\textbf{u})$ geometrically", use what I just said at the end of the above paragraph. As for the special case, what will it look like when $\textbf{u} = (0,0)$?

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