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Math Help - transpose

  1. #1
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    transpose

    Let B be a fixed 2X2 matrix and define a function T:M_2x2 --->M_2x2 by T(A)= AB-BA. Prove or disprove that T is a linear operator.
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  2. #2
    Junior Member roy_zhang's Avatar
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    Quote Originally Posted by natewalker205 View Post
    Let B be a fixed 2X2 matrix and define a function T:M_2x2 --->M_2x2 by T(A)= AB-BA. Prove or disprove that T is a linear operator.
    In order to prove or disprove this statement, we need to know what does it mean that a mapping (or function) is linear. By definition, a transformation (or mapping) T is linear if:
    (i) T(u+v)=T(u)+T(v) for all u,\;v in the domain of T; and
    (ii) T(\mathit{c}u)=\mathit{c}T(u) for all u and all scalar \mathit{c}.

    So in order to prove T is a linear operator, we have to show that it satisfies the above two properties. Let's verify the first property:
    Let U and V be two 2\times 2 matrices and B is the given fixed 2\times 2 matrix, by the given function and basic matrix operations, we have

     T(U+V)=(U+V)B-B(U+V)=UB+VB-BU-BV
    =UB-BU+VB-BV=T(V)+T(U)

    which shows that T satisfies the first property. Next you need to show T(\mathit{c}U)=\mathit{c}T(U) for any given 2\times 2 matrix U and scalar \mathit{c}. Can you pick up from here?

    Roy
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