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Matrices and Linear spaces

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  1. Apr 9, 2017 #1
    1. The problem statement, all variables and given/known data
    The vectors ##a_1, a_2, a_3, b_1, b_2, b_3## are given below
    $$\ a_1 = (3~ 2~ 1 ~0) ~~a_2 = (1~ 1~ 0~ 0) ~~ a_3 = (0~ 0~ 1~ 0)~~ b_1 = (3~ 2~ 0~ 2)~~ b_2 = (2 ~2~ 0~ 1)~~ b_3 = (1~ 1~ 0~ 1) $$
    The subspace of ## \mathbb R^4 ## spanned by ##a_1, a_2, a_3## is denoted by ##V_1## and the subspace of ## \mathbb R^4 ## spanned by ##b_1, b_2, b_3## is denoted by ##V_2##

    The set of vectors which consists a zero vector and all vectors which belong to ##V_1## and ##V_2## is denoted by ##W##.

    Write down two linearly independent vectors which belong to ##W##

    3. The attempt at a solution
    Can somebody please explain how to get the independent vectors? I am very aware of the definition of independent vectors. However, I cannot seem to relate the definition and use it to solve this question.
     
  2. jcsd
  3. Apr 9, 2017 #2

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    Hint: a1, a2, and a3 all have 0 as their 4'th coordinate, so they can not generate any vector with a nonzero 4'th coordinate.
     
  4. Apr 9, 2017 #3
    And ##b_1, b_2, b_3## all have 0 as their 3rd coordinate, so they cannot generate any vector with a nonzero 3rd coordinate
    Thus the resulting vector should be ##(p~q~0~0)##
    Correct?
     
  5. Apr 9, 2017 #4

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    You need to find two vectors. Look for one in V2 that could not be a linear combination of the ais and look for a vector in V1 that could not be a linear combination of the bis
     
  6. Apr 9, 2017 #5
    a3 and b3?
    And the reasoning behind the special criteria for choosing the vector is this right? If I were to choose a vector in V1 which was a linear combination of V2 vectors, then I would technically be choosing just V2 vectors and not taking into account the V1 vectors.
     
  7. Apr 9, 2017 #6

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    In homework, you would want to briefly explain why the two you pick are in W and are linearly independent. Use the definition of linearly independent.
     
  8. Apr 9, 2017 #7
    Thank you very much for your help
     
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