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Fundamental subspace theorem

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.

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.

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