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I Linear Transformation notation

  1. Mar 1, 2016 #1
    I'm confused about the notation
    [tex]
    T:R^n \implies R^m
    [/tex]
    specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
     
  2. jcsd
  3. Mar 1, 2016 #2

    FactChecker

    User Avatar
    Science Advisor
    Gold Member

    That is very possible. It is easy to define linear transformations that go from Rn into a subspace of Rm, where m can be smaller, equal, or greater than n:

    m smaller than n: (x1, x2) => x1
    m equals n: (x1, x2) => (-x1, x1+x2)
    m greater than n: (x1, x2) => (x1, x2, x1+x2); another is (x1, x2) => (x1, 0, 0)
     
  4. Mar 1, 2016 #3

    Mark44

    Staff: Mentor

    That's not the way to think about it. T isn't transforming a single number, like n. It's a map between a space of n dimensions to another space of m dimensions. If n = 2 and m = 3, T is a map from vectors in the plane to vectors in space (three dimensions).

    Ordinary functions, which you're probably more familiar with, are maps from ##\mathbb{R}^1## to ##\mathbb{R}^1##. (I added the 1 exponents only for emphasis.) A function of two variables is a map from ##\mathbb{R}^2## to ##\mathbb{R}^1##.
     
  5. Mar 1, 2016 #4
    Okay makes sense. We're transforming vectors.
    I don't feel confident with my understanding of this statement. What does it mean from vectors in the plane
     
  6. Mar 1, 2016 #5

    Mark44

    Staff: Mentor

    Two-dimensional vectors, like <2, -1>. A vector in space would be, for example, <3, 1, 2>.
     
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