You are told that the linear transfromation is "from to " so you will be multiplying a matrix by a "column matrix" with 2 entries (actually 1 column, 2 rows as a matrix) so there must be two columns in you matrix to be able to multiply. The result is in [itex]R^3[/itex] so it will be a "column matrix" with 3 entries (1 column, 3 rows) which means your matrix must have 3 rows to give 3 results. The matrix representing this linear transformation must be "2 by 3" (2 columns, 3 rows).
Do as Ackbeet suggested: since you titled this "Standard matrices and basis vectors" he, and I, assume that and .
You are told that so what is T(1, 0)? What is T(0, 1)?
Now use the fact that and to find a, b, c, d, e, and f.