# Thread: Geometric description of span

1. ## Geometric description of span

never mind ignore this post

2. ## Re: Geometric description of span

Originally Posted by otownsend
"Span{v1, v2} is the set of all points on the line through v1 and 0".
I think your given answer is wrong. Rather, I would say that
Span{v1} is the set of all points on the line through v1 and 0
and
Span{v1, v2} is the set of all points on the plane through v1, v2 and 0

3. ## Re: Geometric description of span

Originally Posted by Archie
I think your given answer is wrong. Rather, I would say that
and
I agree with otownsend's description.

otownsend, are you familiar with scalar multiplication of vectors? That is, $\vec{v} = <i,j,k>$, given any real number $a$, we have $a\vec{v} = <ai,aj,ak>$.
With respect to your vectors, you have $\vec{v_2} = \dfrac{3}{2}\vec{v_1}$. Because $\vec{v_2}$ is a scalar multiple of $v_1$, it is along the same line as the line between 0 and $v_1$. Therefore, it is in the same span.

4. ## Re: Geometric description of span

In the case that v_1 and v_2 are colinear (which the OP denied, and I didn't check), I agree.

ignore this

6. ## Re: Geometric description of span

Would this also suggest that the span is within R3 ?

7. ## Re: Geometric description of span

Originally Posted by SlipEternal
I agree with otownsend's description.

otownsend, are you familiar with scalar multiplication of vectors? That is, $\vec{v} = <i,j,k>$, given any real number $a$, we have $a\vec{v} = <ai,aj,ak>$.
With respect to your vectors, you have $\vec{v_2} = \dfrac{3}{2}\vec{v_1}$. Because $\vec{v_2}$ is a scalar multiple of $v_1$, it is along the same line as the line between 0 and $v_1$. Therefore, it is in the same span.
would this also suggest that the span is within R3 ?

8. ## Re: Geometric description of span

Originally Posted by otownsend
would this also suggest that the span is within R3 ?
What do you mean "within"? If you take the span of any number of vectors in $\mathbb{R}^3$, the span of those vectors will always contains only vectors in $\mathbb{R}^3$. For example, the vector $\hat{i} = <1,0,0>$ is a vector in $\mathbb{R}^3$. Its span is given by $\{x\hat{i} | x \in \mathbb{R} \} \cong \mathbb{R}$. Basically, you have a one dimensional subspace of $\mathbb{R}^3$, but every vector is still a vector in three dimensions.

9. ## Re: Geometric description of span

ok gotcha thanks.