Use the steps below to prove the following relations among

Chapter , Problem 13E

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QUESTION:

Problem 13E

Use the steps below to prove the following relations among the four fundamental subspaces determined by an m × n matrix A.

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QUESTION:

Problem 13E

Use the steps below to prove the following relations among the four fundamental subspaces determined by an m × n matrix A.

ANSWER:

Solution 13E

Step 1 of 3

Consider a  matrix

a)

Suppose  is in Row  and  is in Nul, which implies

This means that each row of A multiplied by is a zero.

So each row of A is orthogonal to in Nul.

By the definition of orthogonal compliment, each row of  will be in

As is a subspace, .

Thus,  is contained in.

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