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    Hey guys i was wondering if you could help me out, heres the question with my working.

    Let M e F . Show that if {v1, v2, . . . , vn} spans V then so does
    {v1 + Mv2, v2, . . . , vn}.

    I know any v ? V can be written as a linear combination of elements of {v1, v2, . . . , vn}, so i applied this and made a substution like so;

    a1 v1 + a2 v2 + ... + an vn = <v1 , ... , vn>

    So if replace v1 with v1 + Mv2, then i get

    a1[v1 + Mv2] + a2 v2 ... + an vn. If i then expand all this i get

    a1v1 + (a1 a2 M)v2 + ... + anvn

    So by defn this spans V.

    Is this correct a simple substution and re-arranging to get it in terms of the defn?

    Thanks for your help.
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    I'm not really convinced by your argument. I think the best way to go about doing this is to write v_1 as a linear combination of vectors from \{ v_1 + Mv_2, v_2, \dots, v_n \} (there's an obvious way to do this), and then since you can get any vector from a set you already know spans V, you must be able to span V with the new set.
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    So you mean write

    V1 = ( x1, x2, x3) + M(y1, y2, y3) = (x1 + My1, x2 + My2, x3 + My3) ?
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    No. If you start with v1, and you add Mv2 to it to get v1+Mv2, then what multiple of v2 could you add to v1+Mv2 to get v1?
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    (Original post by DDave)
    So you mean write

    V1 = ( x1, x2, x3) + M(y1, y2, y3) = (x1 + My1, x2 + My2, x3 + My3) ?
    What are x_i and y_j?

    What I was saying is this:

    For simplicity let S = \{ v_1, v_2, \dots, v_n \}. We know that S spans V.

    Let w = v_1 + Mv_2 so that your new set is S_{\text{new}} = \{ w, v_2, \dots, v_n \}. How can you write v_1 as a linear combination of vectors in S_{\text{new}}? Once you've done that, it means that any vector of S can be written as a linear combination of vectors in S_{\text{new}}. But we know that any vector in V can be written as a linear combination of elements of S, so this means that any vector in V can be written as a linear combination of linear combinations of elements in S_{\text{new}}... but a linear combination of linear combinations is itself a linear combination, so you're done.
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    (Original post by DFranklin)
    No. If you start with v1, and you add Mv2 to it to get v1+Mv2, then what multiple of v2 could you add to v1+Mv2 to get v1?
    Well if you added -Mv2 you would get v1, but i dont understand how that helps?
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    (Original post by DDave)
    Well if you added -Mv2 you would get v1, but i dont understand how that helps?
    Well if you can write v1 as a linear combination of the vectors in Snew, and you still have v2, v3,...vn, then you know you can get all of the vectors in original set S from linear combinations of Snew.

    Once you have all the vectors in S as linear combinations of Snew, you're done, since we already have that S spans V.
 
 
 
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