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Romero

  • 2 years ago

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?

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  1. Zarkon
    • 2 years ago
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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. glgan1
    • 2 years ago
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    I think should be all the combinations.

  3. Romero
    • 2 years ago
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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?

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

  5. Zarkon
    • 2 years ago
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    *of

  6. Romero
    • 2 years ago
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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?

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

  8. Zarkon
    • 2 years ago
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    yes

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

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

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

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

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

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

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

  16. Zarkon
    • 2 years ago
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    yes

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

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

  19. Zarkon
    • 2 years ago
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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

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