Respuesta :
Answer:
True. See the explanation and proof below.
Step-by-step explanation:
For this case we need to remeber the definition of linear transformation.
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 [tex] x,y \in V[/tex] and for all scalars [tex] c \in R[/tex]. And A is called the domain and B the codomain of T.
Proof
For this case the tranformation proposed is t: [tex] M_{mxn} (R) > M_{nxm} (R) [/tex]
Where [tex] T(A) = A^T [/tex]
For this case we have the following assumption:
1) The transpose of an nxm matrix is an nxm matrix
And the following conditions:
2) [tex]T(A+B) = (A+B)^T = A^T + B^T = T(A) + T(B)[/tex]
And we can express like this [tex] T(A+B) =T(A) + T(B)[/tex]
3) If [tex] A \in M_{mxn}(R)[/tex] and [tex] c \in R[/tex] then we have this:
[tex] T(cA) = (cA)^T = cA^T = cT(A)[/tex]
And since we have all the conditions satisfied, we can conclude that T is a linear transformation on this case.
