Are we to assume that u and v are basis vectors for this space?

If not, then I have no idea what something like "T1(v) = 7v - 7u" could mean. If so, set the basis vectors in a specific order, say first u, then v.

Apply the linear transformation to u and v in order:

T1(u)= 5u -6v (notice that because I decided on the order "first u, then v" I wrote u first v second.) That is a linear combination of the basis vectors with coefficients 5 and -6. That is the first column of our matrix.

T1(v)= -7u+ 7v. The second column is the coeffcients -7 and 7. The matrix representing T1 in this ordered basis is

Do you see why that works? u itself is 1u+ 0v and so would be represented, in this ordered basis, by . Multiplying any two column matrix by that would give exactly the first column. On the other hand, v= 0u+ 1v and so would be represented by . Multiplying that by any matrix gives the second column.