Question

How to construct a matrix when column space and null space are given?

Answer 1

Construct a 4 x 4 matrix A whose column space R and null space N are given by

R=α⎡⎣⎢⎢⎢1200⎤⎦⎥⎥⎥+β⎡⎣⎢⎢⎢0120⎤⎦⎥⎥⎥R=α[1200]+β[0120]

N=α⎡⎣⎢⎢⎢1200⎤⎦⎥⎥⎥+β⎡⎣⎢⎢⎢0120⎤⎦⎥⎥⎥N=α[1200]+β[0120]

How do I approach this problem?

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askedNov 22 '16 at 4:49



flashburn
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3 Answers

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Let A=[v1v2v3v4]A=[v1v2v3v4] be the required matrix. You need that

A⎡⎣⎢⎢⎢1200⎤⎦⎥⎥⎥=v1+2v2=0A[1200]=v1+2v2=0

so v1=−2v2v1=−2v2; also

A⎡⎣⎢⎢⎢0120⎤⎦⎥⎥⎥=v2+2v3=0A[0120]=v2+2v3=0

so v2=−2v3v2=−2v3 and v1=4v3v1=4v3.

You see that v4v4 can be anything; now just take

v3=⎡⎣⎢⎢⎢1200⎤⎦⎥⎥⎥v4=⎡⎣⎢⎢⎢0120⎤⎦⎥⎥⎥v3=[1200]v4=[0120]

Then the matrix

A=[v1v2v3v4]=⎡⎣⎢⎢⎢4800−2−40012000120⎤⎦⎥⎥⎥A=[v1v2v3v4]=[4−2108−42100020000]

will have the required column space and null space: indeed, the column space is generated by v3v3and v4v4, so it has dimension 22. Since the null space contains the required one, it will be equal to it by the rank-nullity theorem.