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Thread: Theorem about subspaces

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    Question Theorem about subspaces

    Hi,

    I hope someone can help. I'm trying to understand the following theorem (about subspaces):

    Theorem about subspaces-screen-shot-2017-11-17-7.40.41-pm.png


    I understand the first property. However the second property doesn't make any sense to me. My professor said that the second property is saying that out of all the sets which have property (1), the span must be the smallest. I don't see how this is communicated in property (2). Could someone provide some clarification on this? If you could provide it in the context of a column space that would be the most helpful since this is a subspace that I am most familiar with.

    - otownsend
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    Re: Theorem about subspaces

    Property (2) says that if W is a space containing all those $\displaystyle v_j$s then S is a subset of or equal to W. That immediately implies that there cannot be a subspace smaller than (i.e. a subset of) S that contains all those $\displaystyle v_j$s. That says that S is the smallest such subspace.
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    Re: Theorem about subspaces

    How could S possibly be a smaller subspace than W, given that they both contain all the vj's? It seems like both properties are saying that they are subspaces of R^n which contain the same vectors. So with that train of thought, it seems that the subspace of S is equal to the subspace of W. How is this not true?
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    Re: Theorem about subspaces

    Quote Originally Posted by otownsend View Post
    How could S possibly be a smaller subspace than W, given that they both contain all the vj's? It seems like both properties are saying that they are subspaces of R^n which contain the same vectors. So with that train of thought, it seems that the subspace of S is equal to the subspace of W. How is this not true?
    W may have other members not belonging to S.

    Someone check me on this... Part (2) says "If $\displaystyle W \subset \mathbb{R} ^n $ is subspace containing the $\displaystyle \vec{v_{j}}$ then $\displaystyle S \subset W$ ."

    Shouldn't that last statement read $\displaystyle S \subseteq W$?

    -Dan
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    Re: Theorem about subspaces

    Let us look at an example

    S = span of one vector {1,0,0} = the x-axis

    W = span of two vectors {1,0,0} and {0,1,0} = the x and y axes (the x-y plane)

    $\displaystyle R^3$ = span of three vectors {1,0,0}, {0,1,0}, and {0,0,1} = 3 dimensional space

    $\displaystyle S\subset W\subset R^3$

    Now W does not equal S even though {1,0,0} is in both S and W
    Last edited by Idea; Nov 19th 2017 at 08:30 AM.
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    Re: Theorem about subspaces

    Yes, it is possible that S is a strict subset of W but that was not topsquark's point. It is also possible that S and W are the same set. The conclusion should be $\displaystyle S\subseteq W $.
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    Re: Theorem about subspaces

    Quote Originally Posted by topsquark View Post
    Shouldn't that last statement read $\displaystyle S \subseteq W$?

    -Dan
    For whatever reason, topologists have adopted $\displaystyle \subset $ to mean subset or equal to and $\displaystyle \subsetneq $ as subset not equal to (or strict subset).
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