Let t: mmxn(r)mmxn(r) be the function defined
byt(a)=at ( the transpoose of a) for a
inmmxn(r). is t a linear transformation? justify your
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 and for all scalars . And A is called the domain and B the codomain of T.

Proof

For this case the tranformation proposed is t:

Where

For this case we have the following assumption:

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

And the following conditions:

2)

And we can express like this

3) If and then we have this:

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

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