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Math Help - Dependent/Combination Vectors

  1. #1
    Super Member Deadstar's Avatar
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    Dependent/Combination Vectors

    1) Let V be a vector space and a,b,c,d \in V. Show that the following vectors are linearly dependent.

    v_1 = a + b + c -44d
    v_2 = a - 3b + 6c
    v_3 = b + d
    v_4 = 9a + 7c + 5d
    v_5 = 23b - c - d

    So is it...

    [ 1 1 0 9 0 ][a] = [0]
    [ 1 -3 1 0 23][b] = [0]
    [ 1 6 0 7 -1][c] = [0]
    [-44 0 1 5 -1][d] = [0]

    ?

    Will that not give a=b=c=d=0, which is not really a lot of use because could that not just be applied to any combination of vectors? I thought about using Gaussian elimination but again thats not gonna work and i dont know how to use it on an m x n (m not equal to n{whats the latex for this?}) matrix. So how do you do it?

    2) Write w as the linear combination of the vectors v_1, v_2, v_3
    w =(2,1,1), v_1 = (1,5,1), v_2 = (0,9,1), v_3 = (-3,3,1)

    Now i can see almost straight away that w = v_3 + v_2 - v_1 but im not sure if the correct method to actually solve these. Is there one? Guassian Elimination?
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  2. #2
    Super Member Deadstar's Avatar
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    actually ive solved part 2 so its just 1 now.
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  3. #3
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    Quote Originally Posted by Deadstar View Post
    1) Let V be a vector space and a,b,c,d \in V. Show that the following vectors are linearly dependent.

    v_1 = a + b + c -44d
    v_2 = a - 3b + 6c
    v_3 = b + d
    v_4 = 9a + 7c + 5d
    v_5 = 23b - c - d

    So is it...

    [ 1 1 0 9 0 ][a] = [0]
    [ 1 -3 1 0 23][b] = [0]
    [ 1 6 0 7 -1][c] = [0]
    [-44 0 1 5 -1][d] = [0]

    ?

    Will that not give a=b=c=d=0, Mr F says: Yes. Therefore the set is dependent.

    which is not really a lot of use Mr F says: yes it is.

    because could that not just be applied to any combination of vectors? Mr F says: Of these vectors, yes. Is it really that surprising that in a vector space V of dimension four, five vectors from V will always be linearly dependent?

    I thought about using Gaussian elimination but again thats not gonna work and i dont know how to use it on an m x n (m not equal to n{whats the latex for this?}) matrix. So how do you do it?

    2) Write w as the linear combination of the vectors v_1, v_2, v_3
    w =(2,1,1), v_1 = (1,5,1), v_2 = (0,9,1), v_3 = (-3,3,1)

    Now i can see almost straight away that w = v_3 + v_2 - v_1 but im not sure if the correct method to actually solve these. Is there one? Guassian Elimination?
    ..
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