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Math Help - Linearly independent question

  1. #1
    Member princess_anna57's Avatar
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    Linearly independent question

    A little confused with this concept, how do you know if a set of vectors is linearly independent or not? Been reading my textbook and it doesn't really make sense to me. So for example I have these questions?

    P = {(2, 1), (1, 2), (3, 3)}
    Q = {(2, 1), (1, 2)}
    S = {(3, 5, 7), (4, 6, 8)}

    How do I know if they're linearly independent or not?
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  2. #2
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    Krizalid's Avatar
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    well, there're some procedures.

    for the first one, we have vectors on \mathbb R^2, and you have three, so we have a linearly dependent set. One way to figure who's the bad guy (the dependent vector) is you to put them all into a matrix, like this:

    \left[ \begin{matrix}<br />
   2 & 1  \\<br />
   1 & 2  \\<br />
   3 & 3  <br />
\end{matrix} \right], now do elementary operations to reduce that matrix into row echelon form. (You'll see that the third row will become into zeros, and that shows the vector who makes your set linearly dependent.) Also note that the sum of the two first vectors yields the third one.

    for the second one, put it into a matrix two and compute its determinant. It suffices to show it's not zero to see the linear independence. (Actually, when working on \mathbb R^n and having n vectors, the determinant is the fastest way.)

    finally, the third one, put'em into a matrix and reduce it into the row echelon form.
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  3. #3
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    mr fantastic's Avatar
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    Quote Originally Posted by princess_anna57 View Post
    A little confused with this concept, how do you know if a set of vectors is linearly independent or not? Been reading my textbook and it doesn't really make sense to me. So for example I have these questions?

    P = {(2, 1), (1, 2), (3, 3)}
    Q = {(2, 1), (1, 2)}


    S = {(3, 5, 7), (4, 6, 8)}

    How do I know if they're linearly independent or not?
    For the set P, assume linear dependence. Under this assumption there will exist scalars a, b and c, not all zero, such that

    a <2, 1> + b <1, 2> + c <3, 3> = 0

    \Rightarrow <2a + b + 3c, a + 2b + 3c> = 0

    Therefore:

    2a + b + 3c = 0 .... (1)

    a + 2b + 3c = 0 .... (2)

    (1) - (2): a - b = 0 \Rightarrow a = b. Therefore a = -c.

    Therefore the vectors are linearly dependent.

    Do the other 2 in a similar way and you will find that all the scalars need to be zero. Therefore the sets Q and S are linearly independent.
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