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## Homework Statement

If col (A) is column space of A and ker(A) null space of A

with ker(A) = {Ax = 0}

and ker(A') = {A'y = 0}

## Homework Equations

Consider the (3x2) matrix :

A = [1,2 ; 3,4 ; 5,6] (matlab syntax)

Show that

col(A) = c1 * [1,0,-1]' + c2 * [0,1,2]'

## The Attempt at a Solution

Find c1 and c2, starting from Col(A) definition.

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We tried in this way :

Col(A) = [c1, c2, -c1+c2]'

Ax=Col(A)

Then:

[1,2 ; 3,4 ; 5,6]*[x1 ; x2]=[c1 ; c2 ; -c1+c2]

rank(A)=2

rank(A|Col(A))=2

and the system is soluble.

With Cramer I have:

x1=c2-2c1

x2=-1/2*(c2-3c1)

Ax=0 for ker(A)

and i do the system

1st eq: c2-2c1=0

2nd eq: c2-3c1=0

But here the solution is meaningless.

How can I solve this exercise?

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