How to define a linear transformation

Question

Let T and U be the linear operators on $R^2$ defined by:

$T(x,y)$=$(y,z)$ and $U(x,y)$=$(y, 0)$

Give rules like the ones defining T and U for each of the transformations (U+T),UT,TU,$T^2,U^2.$

I would say that U+T is defined by:

(U+T)(x,y)= (2y+x)

But I do now know how would I define the others

Answer 1

If $T(x,y) = (y,x), U(x,y) = (y,0)$ (though that is not what is posted in the question...)

$(U+T)(x,y) = U(x,y) + T(x,y) = (y,x) + (y,0) = (2y,x)$

$UT(x,y) = U(T(x,y)) = U(y,x) = (x,0)\\ T^2(x,y) = T(T(x,y)) = T(y,x) = (x,y)$

I will leave $UT(x,y), U^2(x,y)$ to you.

Alternatively,

$T = \begin{bmatrix} 0&1\\1&0\end{bmatrix}\\ U = \begin{bmatrix} 0&1\\0&0\end{bmatrix}$

Answer 2

Since any linear transformation can be on a finite-dimensional vector space can be expressed using a matrix, the best would be to express $T$ and $U$ as matrices.

All your composed transformations can then simply be expressed as matrix operations. Adding two transformations is adding the matrices and compositions becomes matrix multiplication.