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Suppose I have $2$ planes in $R^3$ and they form a system $Ax=b$.

I know the NullSpace of $A$ represents geometrically the vectors that form the intersection between the 2 planes shifted to the origin.

I also know that the Row Space of A represents the span of the normal vectors to the 2 planes.

But i was looking for some geometrical meaning for the Column Space of $A$.
They contain all the vectors $b$ that make $Ax=b$ have at least one solution, that i know.

What I'm trying to see is what does the vectors from the Column Space mean in relating to the $2$ planes in $R^3$ or relating to the possible shiftings the system of planes could suffer to still yield a solution.

nerdy
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1 Answers1

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Well, they live in a separate, 'transpose' world. Now, as you have $2$ equations (of planes), $b$ and the columns will be elements of $\Bbb R^2$ and not $\Bbb R^3$.

I doubt they would represent anything special, related to the $2$ planes in $\Bbb R^3$, a strong reason is that if we multiply the equation of a plane (a row) by a scalar, we get the same plane, but the column space suffers a radical change..

Berci
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  • Maybe they don't mean anything directly to the planes but the possibitilies of shifting the planes from the origin to some point where we could yield at least 1 solution ? That's the kind of answear that i'm looking for – nerdy Apr 06 '13 at 11:20