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`Forgotten' linear algebra

  1. Mar 28, 2008 #1
    Hi all,

    I learned this stuff years ago and wasn't brilliant at it even then so I think a refresher is in order.

    Suppose I have n distinct homogeneous equations in n unknowns. I want to find the solution so I write down the matrix of coefficients multiplying my vector of variables as follows

    [itex]A \mathbf{x} =\mathbf{0}[/itex].

    Now, we don't want [itex]\deta A \neq 0[/itex] to happen otherwise the columns of A are linearly independent so the only solution to [itex]A \mathbf{x} = \mathbf{C}_1 x_1 + \cdots \mathbf{C}_n x_n = \mathbf{0}[/itex] is [itex]\mathbf{0}[/itex].

    Now how do we actually solve this for [itex]\mathbf{x}[/itex], do we just do Gaussian elimination followed by back-substitution? Is the solution unique in this case?

    Now suppose the system is inhomogeneous

    [itex]A\mathbf{x} = \mathbf{b}[/itex] where [itex]\mathbf{b}\neq 0[/itex]. In this case we actually want [itex]\det A \neq 0[/itex] because then we can instantly write down the unique solution

    [itex]\mathbf{x} = A^{-1}\mathbf{b}[/itex].

    Have I gotten the solution to square systems about right? If yes, I'll try to figure out the non-square case.
  2. jcsd
  3. Mar 28, 2008 #2
    If the null space of A is not empty, then it is a subspace, so the solution is an entire subspace of the space you're working with, not just a single vector. The subspace containing only the zero vector is the only degenerate subspace that does consist of a single vector, and it is always in the null space.

    Yep, that's right.
    Last edited: Mar 28, 2008
  4. Mar 28, 2008 #3
    i suggest that after you write your matrix
    just make a row reduction
    and you are not supposed to write a column of zeros in the end
    the last column depends on the last number after the "=" sign
  5. Mar 28, 2008 #4
    You can also solve systems of equations of this form with the wedge product (wedging the column vectors). I'd put an example of this in the wiki Geometric Algebra page a while back when I started learning the subject:


    Looking at the example now, I don't think it's the greatest. It should also probably be in a wedge product page instead of GA ... but that was the context that I learned about it first (I chose to use the mostly empty wiki page to dump down my initial notes on the subject as I started learning it;)
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