Question 4.1.6: Define a mapping T: Mm×n → Mn×m by T (A) = A^t Show that the...

Define a mapping T: M_{m\times n} → M_{n\times m} by

 T (A) =A^{t}

Show that the mapping is a linear transformation.

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By Theorem 6 of Sec. 1.3, we have

T (A+ B) = \left(A+ B\right) ^{t}= A^{t} + B^{t}= T (A) + T (B)

Also by this same theorem,

T (cA) =\left(cA\right) ^{t} = cA^{t} = cT (A)

Thus, T is a linear transformation.

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