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Thread: Geometric description of span

  1. #1
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    Question Geometric description of span

    never mind ignore this post
    Last edited by otownsend; Oct 3rd 2017 at 09:21 AM.
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    Re: Geometric description of span

    Quote Originally Posted by otownsend View Post
    "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
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    Re: Geometric description of span

    Quote Originally Posted by Archie View Post
    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.
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    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.
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    Question Span and linear independence

    ignore this
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    Re: Geometric description of span

    Would this also suggest that the span is within R3 ?
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    Re: Geometric description of span

    Quote Originally Posted by SlipEternal View Post
    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 ?
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    Re: Geometric description of span

    Quote Originally Posted by otownsend View Post
    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.
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    Re: Geometric description of span

    ok gotcha thanks.
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