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Math Help - Spanning and linear independence

  1. #1
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    Spanning and linear independence

    Let v1,v2,...vn be linearly independent vectors in a vector space V. Show that v2,...vn cannot span V.

    I'm confused because doesn't the linear independency proof contain the fact that say vectors v1,...vn are contained in Span(v1,...vn)? And then it is written uniquely as a linear combination of v1,...vn if and only if v1,...vn are linearly independent.
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  2. #2
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    Quote Originally Posted by pakman View Post
    Let v1,v2,...vn be linearly independent vectors in a vector space V. Show that v2,...vn cannot span V.

    I'm confused because doesn't the linear independency proof contain the fact that say vectors v1,...vn are contained in Span(v1,...vn)? And then it is written uniquely as a linear combination of v1,...vn if and only if v1,...vn are linearly independent.
    If v_2,...,v_n spam V then this set form a basis for V. We claim that v_1,v_2,...,v_n cannot be linearly independent. Because since v_2,...,v_n spam it means for any vector v we can express it as a linear combination of those. Thus, v_1 = k_2v_2+...+k_nv_n. Thus, v_1 - k_2v_2 -... - k_nv_n = 0 not all the coefficients are zero, so it cannot be linearly independent.
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