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Matrix vector product and linear transformation proof

  1. Sep 24, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]A(\vec{a}+\vec{b}) = A\vec{a} + A\vec{b}[/tex]
    Where A is a matrix and T (in the following section) is a transformation.

    2. Relevant equations
    [tex]T(\vec{a}) + T(\vec{b}) = T(\vec{a}+\vec{b})[/tex]
    [tex]T(\vec{a}) = A\vec{a}[/tex]
    [tex]T(\vec{b}) = A\vec{b}[/tex]

    3. The attempt at a solution
    If [tex]\vec{a}+\vec{b} = \vec{c}[/tex]
    [tex]T(\vec{a}+\vec{b}) = T(\vec{c}) = Ac = A(\vec{a}+\vec{b})[/tex]
    [tex]T(\vec{a}+\vec{b}) = A(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} + A\vec{b}[/tex]

    Is this a sufficient proof? I can do it the more arduous way, but I think this is a proof, isn’t it?
    Any help appreciated.
  2. jcsd
  3. Sep 24, 2010 #2


    Staff: Mentor

    I assume you are using the properties of a linear transformation. I would do it this way.

    [tex]A(\vec{a} + \vec{b}) = T(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} +A\vec{b}[/tex]
  4. Sep 24, 2010 #3
    Thanks Mark44.
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