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Linear algebra

  1. Aug 5, 2013 #1
    The matrix is an example of a Linear Transformation, because it takes one vector and turns it into another in a "linear" way.

    Hi could you explain it what exactly the bold part suggest?

    What is "another in a "linear" way"?
     
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  3. Aug 5, 2013 #2

    WannabeNewton

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    Actually any linear map between two finite dimensional vector spaces (over the same field) can be represented as a matrix so matrices aren't exactly "special" in that sense. A linear map ##L: V \rightarrow W## sends vectors in ##V## to vectors in ##W## such that ##L## preserves addition and scalar multiplication from one vector space to the other.
     
  4. Aug 5, 2013 #3

    Fredrik

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    A linear transformation is a map ##f:U\to V## such that U and V are vector spaces over the same field F (typically, ##F=\mathbb R## or ##F=\mathbb C##), and
    $$f(ax+by)=af(x)+bf(y)$$ for all ##x,y\in X## and all ##a,b\in F##. If A is an n×n matrix, then the map ##x\mapsto Ax## is linear, because
    $$A(ax+by)=aAx+bAy$$ for all n×1 matrices x,y and all real (or complex) numbers a,b.

    Note that we can define a function f by defining f(x)=Ax for all n×1 matrices x. Since both the domain and codomain of f is a set whose elements are n×1 matrices to n×1 matrices, it can be said to "take a vector and turn it into another vector". When they say that it does so "in a linear way", they mean that the function is linear in the sense defined above.
     
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