Let A and B be vector spaces with same scalars. A map defined as T: A >B is called a linear transformation from A to B if satisfy these two conditions:

1) T(x+y) = T(x) + T(y)

2) T(cv) = cT(v)

For all vectors $x,y \in V$ and for all scalars $c \in R$ . And A is called the domain and B the codomain of T.

Proof

For this case the tranformation proposed is t: $M_{mxn} (R) > M_{nxm} (R)$

Where $T(A) = A^T$

For this case we have the following assumption:

1) The transpose of an nxm matrix is an nxm matrix

And the following conditions:

2) $T(A+B) = (A+B)^T = A^T + B^T = T(A) + T(B)$

And we can express like this $T(A+B) =T(A) + T(B)$

3) If $A \in M_{mxn}(R)$ and $c \in R$ then we have this:

$T(cA) = (cA)^T = cA^T = cT(A)$

And since we have all the conditions satisfied, we can conclude that T is a linear transformation on this case.

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3 years ago