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Math Help - Linear independence

  1. #1
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    Linear independence

    Let u,v be distinct vectors in a vector space V. Show that {u,v} is linearly
    dependent if and only if u or v is a multiple of the other.
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  2. #2
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    Quote Originally Posted by JCIR View Post
    Let u,v be distinct vectors in a vector space V. Show that {u,v} is linearly
    dependent if and only if u or v is a multiple of the other.
    So we have a statement

    p \iff q

    where p is if u,v are linearly dependent and q is if u or v is a multiple of the other

    Since we are proving and if and only if statement we need to prove both directions. So lets start with

    p \implies q

    We assume that u and v are linearly dependant. So by definition there exits scalers c_1,c_2 \in \mathbb{R} such that c_1,c_2 \ne 0 and c_1u+c_2v=0 Now if we solve this equation for u we get u=-\frac{c_2}{c_1}v. Therefore u and v are multiples of each other. Done.

    Now for the other direction

    q \implies p

    We assume that there are mulitples of each other so we get

    v=ku where k \ne 0Now we subract v from both sides to get

    0=ku-1\cdot v Now we have to non zero scalers and a linear combination that is equal to zero. So the vectors are dependant.

    QED.

    I hope this helps.
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  3. #3
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    Yes it helps greatly.
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