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Linear algebra question (span?)

  1. Sep 20, 2009 #1

    ckp

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    How would i go about telling if vector b (4 row, 1 column) is in the span of the columns of matrix A(4 row, 5 column)?

    im just not sure what is asking, i know it would be an easy question if i knew what they meant by this.
     
  2. jcsd
  3. Sep 20, 2009 #2
    Same question reformulated: does the system Ax=b have a solution.
     
  4. Sep 20, 2009 #3

    ckp

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    so would it not have a solution because there are 4 eq and 5 unknowns? also how would i justify in terms of span that there is no solution? (as opposed to just saying because there are 4 eq and 5 unknowns)
     
  5. Sep 20, 2009 #4
    Do you know the actual matrix A and the vector b? Form the augmented matrix [A|b] and row reduce to see if the system is consistent or not.
     
  6. Sep 20, 2009 #5

    ckp

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    Is this possible with A being 4rows x 5 columns and b 4 rows 1 column?
     
  7. Sep 20, 2009 #6
    Maybe yes:

    1 0 0 0 0 | 1
    0 1 2 0 0 | 1
    0 0 0 1 0 | 1
    0 0 0 0 1 | 1

    Maybe not:

    1 0 0 0 0 | 1
    0 1 2 0 0 | 1
    0 0 0 1 0 | 1
    0 0 0 0 0 | 1

    Depends on A and b
     
  8. Sep 20, 2009 #7

    ckp

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    So, say it is consistent. Then what do I do? (rr took a long time and numbers are rather large)
     
  9. Sep 20, 2009 #8
    If the system Ax=b is consistent, then the answer is "yes, b is in the span of the columns of A."
     
  10. Sep 20, 2009 #9

    ckp

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    now, what if it is asked if a sub n is in the span of A (A consists of {a sub 1,...,a sub n})
     
  11. Sep 21, 2009 #10
    Are you asking this: "is the rightmost column of A in the span of the columns of A?"

    ??
     
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