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So I have a question that says...

Find two nonzero vectors u and v in $R^2$ such that span(u,v) doesn't = $R^2$

I've been trying to figure this out, but I'm so confused. I don't get how to vectors can be in $R^2$, but there span isn't. I know the span is the set of all linear combinations, which is just multiplying the vectors by a scalar value so that the vector becomes something like $c1v1+c2v2+...+ckvk$ where vectors $v1,v2,...vk $ are in $R^2$ and $c1,c2,...ck$ are scalars. So how does the scalar values change the vectors so that there not in $R^2$ anymore? Can someone explain this problem to me? Is there a concept about spans that I'm missing/misunderstanding?

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  • $\begingroup$ What if both vectors lie along the same line? $\endgroup$ – Evan Aad Sep 25 '16 at 4:24
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I think you're misunderstanding the concept of span. If we say "the set $\{u_1,u_2\}$ (which are in $R^2$) spans $R^2$" we mean that every vector in $R^2$ can be written as a linear combination of these two vectors. When we say that a set $\{v_1,v_2\}$ (vectors in $R^2$) does not spans $R^2$ it means that there are some vectors on $R^2$ which can't be written as a linear combination of the set $\{v_1,v_2\}$. It doesn't mean that $\{v_1,v_2\}$ generates vectors outside $R^2$, but means that the set $\{v_1,v_2\}$ actually doesn't generates all the vectors in $R^2$. In this case we say that the set $\{v_1,v_2\}$ is "linearly dependent", because although they are two distinct vectors, we can't write every element on $R^2$ as a linear combination of $\{v_1,v_2\}$. Geometrically speaking, two linearly dependent vectors of $R^2$ are those which one of them can be obtained as a scalar multiplication of the other. That is, they are parallel vectors, which clearly can't span all $R^2$, but only a straight line spanned by one of the vectors: $v_1$ or $v_2$ (because they are parallel).

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Multiplying a vector in $\mathbb R^2$ by a real scalar will still result in a vector in $\mathbb R^2$. What the question is asking is for a pair of vectors whose span is a part of $\mathbb R^2$, but not all of $\mathbb R^2$. What relationship must these two vectors have to ensure that they don't span all of $\mathbb R^2$?

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Because their span is a line, for example $(1,2)$ and $(2,4)$, both in it but they only span a line.

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Suppose that $\mathbf u$ and $\mathbf v$ are either parallel or antiparallel i.e. Suppose that $\mathbf v = k\mathbf u$ for some arbitrary choice of $k$ their span is not $\mathbb R^2$.

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