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Math Help - The interpretation of the null-space

  1. #1
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    The interpretation of the null-space

    Let A denote an mxn matrix and let A' denote the row echelon form of it, which has d steps. We then have according to my textbook:
    1) If m>d there exists a column such that the set of equation has no solutions.
    2) If n>d the matrixequation AX=0 has a set of solution expressed parametrically by (n-d) parameters.
    3) If m=n=d there exists a unique solution for every B in the equation AX = B.

    Now 1) and 3) I understand. What troubles me is 2). Why to they switch the equation to AX=0 rather than AX=B? Wouldn't that last equation also be dependent on n-d parameters. I'm pretty sure that this has something to do with the fact that the nullspace is quite a unique thing since it forms a linear subspace. Thus we can define its dimension and later use all this to prove the rank nullity theorem. So can someone explain what's going on on a deeper level?
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  2. #2
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    Re: The interpretation of the null-space

    ok, when we solve a system of linear equations, which we can write in short-hand form as AX = B, one thing that is often done is to look at the homogeneous system AX = 0 first. why? because knowing the dimension of the null space {X in R^n: AX = 0}, tells us some qualitative things about the solution set {X in R^n: AX = B}.

    in particular, if X1 is any specific solution to AX = B, and X2 is in the null space, then A(X1 + X2) = A(X1) + A(X2) = B + 0 = B. so if we want to find ALL the solutions to AX = B, we need to find ONE solution for AX = B, and ALL the solutions to AX = 0. often this is summed up as "general solution = particular solution + homogeneous solutions".

    when the rank of the matrix A (i am presuming this is what they mean by "A' has d steps") is less than the dimension of the co-domain of A (R^m), there are going to be vectors B in R^m that aren't in the image of A. when the rank is less the the dimension of the domain of A (R^n) (often expressed as "fewer equations than unknowns", although this is not very accurate), the "n-d parameters" are essentially what we need to specify to include all the solutions that the null space of A gives us (since dim(null(A)) = n-d), over and above any particular solution we find.

    for example, consider the following system:

    x + 3z = 1
    y + 4z = 2

    the matrix A =

    [1 0 3]
    [0 1 4]

    which is already in reduced row-echelon form. the null space of A is all vectors in R^3 of the form: (-3t,-4t,t) = t(-3,-4,1), which has the basis {(-3,-4,1)}. now to find a particular solution, we could let z = 0 (which is convenient), so (1,2,0) is a particular solution. then the general solution is: (1,2,0) + t(-3,-4,1).

    in geometric terms, the solution set (of AX = B) is just the line that makes up the null space of A, "translated" by a particular vector in the solution set.
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  3. #3
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    Re: The interpretation of the null-space

    Okay that helped a lot. But let's just assume you have a reduced row echelon form of an mxn matrix A with rank d such that m>d. Would you then always be guaranteed that the set of solutions to not only the system of equations AX=0 but also generally the system AX=B would be expressed through n-d parameters? Or could that vary depending on what B you choose?
    I do realize that this is quite irrelevant since the solutions to AX=0 are more interesting but I still want to know.
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