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Thread: Please help me understand a property of a matrix

  1. #1 Please help me understand a property of a matrix 
    Forum Freshman
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    Let A be a 3 by 4 matrix.

    A =

    1 -4 8 1
    0 2 -1 3
    0 0 0 5

    My textbook says that in order for the columns of A to span Rm, A must have a pivot position in every row. We see that it does.

    My problem is that I thought that the matrix

    1 -4 8 1
    0 2 -1 3
    0 0 0 0
    0 0 0 5

    meant the exact same thing as the matrix A. But then it would not span Rm because there is a row with no pivot position. What am I not understanding about matrices?


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  3. #2  
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    Compare determinants of your 2 matrices. Are they the same?


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  4. #3  
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    The first matrix maps R4 into R3, the second maps R4 into R4. However, the third coordinate of the second map is 0, so there is a 1-1 correspondence between the images of the two maps.
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  5. #4  
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    Quote Originally Posted by Guitarist View Post
    Compare determinants of your 2 matrices. Are they the same?
    We haven't covered "determinants" yet.


    Another reason why I don't understand the difference between the two matrices is that the solution set would seem to be the exact same for Ax = b.
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  6. #5  
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    Okay, thanks for the help Guitarist and mathman!
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  7. #6  
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    1. If any row or column of a matrix is all zeros, then the determinant is zero - as you can easily check

    2. If the determinant of a matrix is zero, then the row (or column) vectors are not linearly independent, and therefore cannot be a spanning set
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