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Understanding Ax=b

  1. Mar 4, 2013 #1
    ^{}This may be a silly question, but if I have an 8x3 matrix X, for example, then the rows of this matrix will span R3 (and will be linearly dependent). When we find the solution to:

    Xw=t

    where t is an 8x1 matrix of t's. Then each row can be represented as

    [itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] = [itex]t[/itex].

    Each row then forms a unique plane in R3, correct? Does the matrix Xw form a plane? I'm learning about Perceptrons, a form of Artificial Neural Network, in which each row of data is classified as either

    [itex]y^{'}[/itex] = +1 or -1 depending on if [itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] > [itex]t[/itex] or [itex]w_{1}[/itex][itex]x_{i1}[/itex]+[itex]w_{2}[/itex][itex]x_{i2}[/itex]+[itex]w_{3}[/itex][itex]x_{i3}[/itex] < [itex]t[/itex].

    The book states that in the above situation, "The perceptron model [in the example above] is linear in its parameters w and x. Because of this, the decision boundary of a perceptron, which is obtained by setting [itex]y^{'}[/itex]=0, is a linear hyperplane that separates the data into two classes, -1 and +1."

    I'm having a really hard time understanding what this quote is trying to say, because I don't see how Xw=0 "forms a hyperplane" in x-space.
     
  2. jcsd
  3. Mar 4, 2013 #2

    chiro

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    Science Advisor

    Hey 3.141592654.

    The answer will depend on the rank of the matrix.

    If you row reduce the matrix and get a consistent system with n non-zero rows then the system will form an n-dimensional place from those vectors.

    It may be a point, line, or n-dimensional plane depending on the above.
     
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