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owlet

  • one year ago

Spanning set & Linearly (In)dependent Question below. I'm stuck. I don't know what's next...

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  1. owlet
    • one year ago
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  2. amistre64
    • one year ago
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    2 vectors define a plane, a 2d object ... can a 2d object touch all of 3d space?

  3. amistre64
    • one year ago
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    and independant set is one in which each vector cannot be defined in terms of other vectors in the set

  4. owlet
    • one year ago
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    no? 2d can't span all 3d space, so it is not a spanning set of R^3? so since c2 needs c1, then it is linearly dependent?

  5. amistre64
    • one year ago
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    since 2,4,0 is not a scaled version of 2,1,3 the vectors are independant.

  6. amistre64
    • one year ago
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    one way to consider it is ... a v1 + b v2 = v3 for all vectors (v3) in R^3 in order to span (to cross over, to reach all points) of R^3, you have to be able to create them from the given conditions ...

  7. amistre64
    • one year ago
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    a(2,1,3) + b(2,4,0) = (x,y,z) 2a + 2b = x a + 4b = y 3a + 0b = z can the vector (0,1,3) be obtained? 2a + 2b = 0; b=-1 a + 4b = 1; 1-4 doest eqaul 1 3a + 0b = 3 ; a = 1

  8. amistre64
    • one year ago
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    think of a piece of paper, you can draw 2 arrows on it that can be combined to reach all points in that paper ... but in order to get off of the paper you need to add a vector that is not contained on the paper ... a vector that points above/below it.

  9. owlet
    • one year ago
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    oh okay... i think i understand it better. two vectors = only spans a plane so if in case i'm given three vectors, there will be a possibility that they can span R^3 right?

  10. amistre64
    • one year ago
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    it requires n, independant vectors, to span R^n ... yes

  11. amistre64
    • one year ago
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    ... at least might need to be included for rigor

  12. amistre64
    • one year ago
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    two independant vectors create a plane yes

  13. owlet
    • one year ago
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    for the linearly independent/dependent, |dw:1443917976763:dw|

  14. owlet
    • one year ago
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    oh i think it makes sense to me now

  15. amistre64
    • one year ago
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    correct enough yes

  16. owlet
    • one year ago
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    okay thank you again! :)

  17. amistre64
    • one year ago
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    youre welcome

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