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Math Help - Please explain Span and Kernel

  1. #1
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    Please explain Span and Kernel

    Would someone please explain what the KERNEL and SPAN of a matrix are in plain english.
    that would be fantastic!
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jmaximum View Post
    Would someone please explain what the KERNEL and SPAN of a matrix are in plain english.
    that would be fantastic!
    Stop thinking of a matrix as a matrix and start thinking of it as a linear mapping between two vector spaces, M: U \rightarrow V. Then the kernel of your linear map (matrix) is everything that is mapped to the zero vector, \{u: uM=0 \in V\}.

    The span, on the other hand, is the image of your linear map. It is everything in V which is mapped to, \{v \mid v=uM \text{ for some } u \in U\}.
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  3. #3
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    I follow you on the kernel, but I am still not getting it for the span. how do you find the span in the first place?
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  4. #4
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    Strictly speaking, there is no such thing as the "span" of a matrix. "Span" only applies to a set of vectors. The span of a set of vectors is the set of all possible linear combinations of those vectors. The "span" of a matrix is really the span of the columns of the vector thought of as vectors.

    For example, the "span" of the matrix \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} is the span of {<1, 3>, <2, 4>}. And since those are independent vectors, the span is all of R^2.

    The "span" of the matrix \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix} is the span of {<1, 2>, <2, 4>} which, because <2, 4>= 2<1, 2>, is just "all multiples of <1, 2>".

    Now suppose <x, y> is any 2-vector. Then \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}x+ 2y \\ 2x+ 4y\end{bmatrix}. Notice that 2x+ 4y= 2(x+ 2y) so that any vector of the form Av, for this matrix A, lies in the "span" of A.

    That means that another way to think of the "span" of a matrix A is as the "range" of the function f(x)= Ax.

    And, of course, the kernel is the set of x such that f(x)= Ax= 0.

    A more common name for the "span" of a matrix is the "column space" or "image" of the matrix.
    Last edited by HallsofIvy; March 6th 2010 at 02:49 PM.
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    wow. you just made what three textbooks and 1 professor could not explain in a simple fashion sound easy!
    thanks a bunch.

    however, the /math you typed in the middle did not resolve to anything, and I could not quite follow it.
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  6. #6
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    Quote Originally Posted by jmaximum View Post
    wow. you just made what three textbooks and 1 professor could not explain in a simple fashion sound easy!
    thanks a bunch.

    however, the /math you typed in the middle did not resolve to anything, and I could not quite follow it.
    I forgot the ending "[/tex]". I have fixed it.
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