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Vectors and linear dependence

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    Show vectors (0,3,1-1) (6,0,5,1) (4,-7,1,3) form a linearly dependent set in "R^4"

    Does this just mean show the vectors are linearly dependent as I know how to do that, its just the R^4 thats confusing me.

    Also which of the following vectors form a basis for R^3?
    (1,0,0) (2,2,0) (3,3,3)

    (3,1,-4) (2,5,6) (1,4,8)

    Is this again just asking which of the vectors are a linearly dependent? I would probably know how to work it out, I just can't get my head around the wording of the question, so I'm not sure what they are asking?
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    Also find the vector v for the basis S=(v1,v2,v3)

    v=(2,-1,3)
    v1=(1,0,0)
    v2=(2,2,0)
    v3=(3,3,3)

    Any help on to approach this question would be appreciated, im not really sure what Im meant to be looking for here. Isn't the vector v already given?
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    (Original post by Gorrilaz)
    Show vectors (0,3,1-1) (6,0,5,1) (4,-7,1,3) form a linearly dependent set in "R^4"

    Does this just mean show the vectors are linearly dependent as I know how to do that, its just the R^4 thats confusing me.

    Yes, you only need to show the vectors are linearly dependent. \mathbb{R}^4 is simply the vector space concerned.

    (Original post by Gorrilaz)
    Also which of the following vectors form a basis for R^3?
    (1,0,0) (2,2,0) (3,3,3)

    (3,1,-4) (2,5,6) (1,4,8)

    Is this again just asking which of the vectors are a linearly dependent? I would probably know how to work it out, I just can't get my head around the wording of the question, so I'm not sure what they are asking?
    No. To show that vectors from a basis for a space, you have to show (a) that they form a linearly independent set and (b) that they span the vector space (\mathbb{R}^3 in this case)

    Hint: since \mathbb{R}^3 is a three dimensional space, there will be only three vectors in the basis
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    (Original post by Gorrilaz)
    Also find the vector v for the basis S=(v1,v2,v3)

    v=(2,-1,3)
    v1=(1,0,0)
    v2=(2,2,0)
    v3=(3,3,3)

    Any help on to approach this question would be appreciated, im not really sure what Im meant to be looking for here. Isn't the vector v already given?
    If (v1, v2, v3 ) is a basis then
    \vec v=\alpha \cdot \vec v_1+\beta \cdot \vec v_2 +\gamma \cdot \vec v_3
    It means 3 scalar equations . Solve these simoultaneously for \alpha, \beta,\gamma

    Another method would be the base transformation.

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