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    Fundamental subspace theorem

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    Consider the matrix

    a=(1 1 2
    1 2 -1
    3 2 1
    5 5 2)

    Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorem holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T).

    Hint: Notice that the fourth row is the sum of the first three rows.

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    https://brainmass.com/math/matrices/fundamental-subspace-theorem-11648

    Solution Preview

    The solution is attached.

    Problem:

    a=(1 1 2
    1 2 -1
    3 2 1
    5 5 2)

    Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorm holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T).

    Hint: Notice that the fourth row is the sum of the first three rows.

    Solution:

    Let us apply the Gaussian Elimination to this matrix. Then, we get O1 A=U, OA=R, where

    O1 =[1 0 0 0
    -1 1 0 0
    -4 1 1 0
    -1 -1 -1 1 ]

    U=[ 1 1 2
    0 1 -3
    0 0 -8
    0 0 0]

    O=[ -1/2 -3/8 5/8 0
    1/2 5/8 -3/8 0
    1/2 -1/8 -1/8 0
    -1 -1 -1 1]
    R=[ 1 0 0
    0 1 0
    0 0 1
    0 0 0]

    To get Matrix U and O1, we apply the Gaussian Elimination to the matrix A I

    A I = [ 1 1 2 1 0 0 0
    1 2 -1 0 1 0 0
    3 2 1 0 0 1 0
    5 5 2 0 0 0 1 ]

    -- [1 1 2 1 0 0 0
    0 1 -3 -1 1 0 0
    0 -1 -5 -3 0 1 0
    0 0 -8 -5 0 0 1 ...

    Solution Summary

    This gives a matrix and then proves that the fundamental subspace theorem holds. Matrix algebra is analyzed.

    $2.19

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