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Romero Group Title

I remember reading that if you have a set of vectors that if one of the vectors is a lin combination of the vectors before it then the set is lin dependent. What I'm confused about is does that vector have to be lin combination of ALL of the previous vectors or just at least one for the set to be lin dependent?

  • 2 years ago
  • 2 years ago

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  1. Zarkon Group Title
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    a combination of one or more though if you can write it is a l.c. of one of the vectors you can write it is a linear combination of all the vectors

    • 2 years ago
  2. glgan1 Group Title
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    I think should be all the combinations.

    • 2 years ago
  3. Romero Group Title
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    Great so if you want to get the basis of the set of vectors you get rid of the linear combination right?

    • 2 years ago
  4. Zarkon Group Title
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    you get rid off was many vectors as needed to make them independent (but no more)

    • 2 years ago
  5. Zarkon Group Title
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    *of

    • 2 years ago
  6. Romero Group Title
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    Ok let's say we have a a set of 3 vectors. We can't determine if it's lin dependent by inspection yet when we reduce it to echelon form we find a free variable so we only have two pivot points but this was done through row reduction using all the rows meaning we added or subtracted row 1 to row 2 and then used row 2 to make the third row a zero row. At that point if we want to have the basis of the set we have to get rid of a vector. Can we get rid of any vector?

    • 2 years ago
  7. Romero Group Title
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    Can we get rid of any vector? @Zarkon This is basically what I'm asking :)

    • 2 years ago
  8. Zarkon Group Title
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    yes

    • 2 years ago
  9. Romero Group Title
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    That's weird. So you can have three different basis for the set of vectors?

    • 2 years ago
  10. Romero Group Title
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    Does this mean that basis is not unique?

    • 2 years ago
  11. Zarkon Group Title
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    correct the basis is not unique...there can be an infinite number of basis's

    • 2 years ago
  12. Romero Group Title
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    Is there a condition where a basis will be unique?

    • 2 years ago
  13. Zarkon Group Title
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    |dw:1336751060283:dw||dw:1336751083709:dw| both a basis for \(\mathbb{R}^2\)

    • 2 years ago
  14. Zarkon Group Title
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    you would have to be pretty restrictive

    • 2 years ago
  15. Romero Group Title
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    What a bais for a vector space? Can that be infinite as well?

    • 2 years ago
  16. Zarkon Group Title
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    yes

    • 2 years ago
  17. Romero Group Title
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    and you can do this by multiplying the basis by a scalar?

    • 2 years ago
  18. Romero Group Title
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    to get infinite basis?

    • 2 years ago
  19. Zarkon Group Title
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    it is really too complicated to talk about here. I'd recommended you read http://en.wikipedia.org/wiki/Banach_space

    • 2 years ago
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