To fully define the linear transformation T : R10, we need to specify its action on each element of R10. In other words, for every vector x = (x1, x2, ..., x10) in R10, we need to determine the corresponding image vector T(x) under the transformation T.

One possible way to define T is by specifying its action on the standard basis vectors of R10. The standard basis vectors of R10 are given by:

e1 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0)
e2 = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0)
e3 = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0)
...
e10 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1)

To define T, we can specify its action on each of these basis vectors. For example, we can define T(e1) = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), T(e2) = (2, 3, 4, 5, 6, 7, 8, 9, 10, 1), T(e3) = (3, 4, 5, 6, 7, 8, 9, 10, 1, 2), and so on.

Once we have specified the action of T on the basis vectors, we can extend it to all vectors in R10 by linearity. That is, for any vector x = (x1, x2, ..., x10) in R10, we can calculate T(x) by taking a linear combination of the images of the basis vectors. For example, if we have defined T(e1) = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), T(e2) = (2, 3, 4, 5, 6, 7, 8, 9, 10, 1), and T(e3) = (3, 4, 5, 6, 7, 8, 9, 10, 1, 2), then for any vector x = (x1, x2, ..., x10) in R10, we can calculate T(x) as:

T(x) = x1 * T(e1) + x2 * T(e2) + x3 * T(e3) + ... + x10 * T(e10)

In summary, to fully define the linear transformation T : R10, we need to specify its action on the standard basis vectors of R10, and then extend it to all vectors in R10 by linearity.