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Thread: Spanning sets

  1. #1
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    Spanning sets

    Determine if each of the following are true or false, and give a justification.

    (a) Let v1, v2 and v3 be vectors in a subspace in V. If { v1, v2} is a spanning set of V, then {v1, v2,v3} must be a spanning set of V.

    On this one I put true because v3 could be a linear combination of v1, v2.

    (b) Let v1, v2,v3 be vectors in subspace V. If {v1, v2,v3} is a spanning set of V, then { v1, v2} must be a spanning set of V.

    Here I put false because if {v1, v2,v3} is linearly dependent, then a vector can't be removed from the set.

    (c) Every spanning set of R2 must contain at least two vectors.

    Here I put false but I don't really know why.. I know if I have an image of a matrix [-s,s]^T then the spanning set would only be a single vector [-1,1]^T...

    Am I on the right track for any of these?
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  2. #2
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    Re: Spanning sets

    Quote Originally Posted by MrJank View Post
    Determine if each of the following are true or false, and give a justification.
    (a) Let v1, v2 and v3 be vectors in a subspace in V. If { v1, v2} is a spanning set of V, then {v1, v2,v3} must be a spanning set of V.
    On this one I put true because v3 could be a linear combination of v1, v2.
    (b) Let v1, v2,v3 be vectors in subspace V. If {v1, v2,v3} is a spanning set of V, then { v1, v2} must be a spanning set of V.
    Here I put false because if {v1, v2,v3} is linearly dependent, then a vector can't be removed from the set.
    (c) Every spanning set of R2 must contain at least two vectors.
    Here I put false but I don't really know why.. I know if I have an image of a matrix [-s,s]^T then the spanning set would only be a single vector [-1,1]^T...Am I on the right track for any of these?
    In part (a) do you mean "in a subspace V?" If so, you are correct.

    In part (c) what do you think the 2 means in $\mathbb{R}^2~?$
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  3. #3
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    Re: Spanning sets

    I thought R2 was the set of vectors in a 2 dimensional space or all 2x1 vectors.
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    Re: Spanning sets

    Quote Originally Posted by MrJank View Post
    I thought R2 was the set of vectors in a 2 dimensional space or all 2x1 vectors.
    Is it possible for one $2\times 1$ vector to generate all vectors in $\mathbb{R}^2~?$
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  5. #5
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    Re: Spanning sets

    When you put it that way, no. So is the spanning set like all the vectors needed to have a linear combination of all the vectors in the given subspace?
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  6. #6
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    Re: Spanning sets

    Quote Originally Posted by MrJank View Post
    When you put it that way, no. So is the spanning set like all the vectors needed to have a linear combination of all the vectors in the given subspace?
    Then how can part (c) be false?
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