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What is the homogeneous problem? What is the purpose of null space of a vector in this context?

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    The homogeneous problem of *what* exactly? Linear differential equations perhaps?2010-09-29
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    @Agusti Roig and @Robin Chapman i got a fantastic answer.2010-09-29

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The answer to what I think you're asking:

The homogeneous problem is: "Given a matrix $A$, find all solutions to the problem $Ax = 0$, where $x$ is a vector of appropriate dimension, and '0' is the vector of all zeros".

The Nullspace of $A$ is precisely the set of all such solutions. There is always at least one solution to the homogeneous problem. Indeed, $x=0$ (the vector of all zeros) is always a solution. In the case where $A$ is a square and invertible matrix, $x=0$ is the only solution. In general, there can be other solutions, and the set of all solutions (the Nullspace) is actually a subspace. To see why, notice that if $x_1$ and $x_2$ both satisfy $Ax=0$, then so does $a_1 x_1 + a_2 x_2$.

I hope this answers your question. Best of luck!

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    very good, thank you.2010-09-29
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    I would call it a homogeneous system of equations...but still, it's alright.2010-09-29
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HINT $\ $ For $\rm A\:$ linear, $\rm\ A\:X_1 = B = A\:X_2 \ \iff\ 0 \:=\: A\:X_1 - A\:X_2 = A\:(X_1-X_2)$

This implies that the general solution of $\rm\ \ \ \:A\:X = B\ $ is the sum of any particular solution plus a solution of the associated "homogeneous" equation $\rm\ A\:X = 0\:$. This property holds true for every linear operator, e.g. for matrices, linear differential equations, linear recurrences, etc.