Let T R2 rightarrow R2 be a linear operator for which the im

Let T: R^2 rightarrow R^2 be a linear operator for which the images of the standard basis vectors for R^2 are T(e_1) = (a, b) and T (e_2) = (c, d). Find T(4, 1). T (4, 1) =

Solution

(4,1) =4.(1, 0) + 1.(0, 1)

=4.e₁ + 1. e₂

_{T(4, 1) = T (}4.e₁ + 1. e₂₎

=4. T(e₁) + 1. T(e₂)

_{= 4. (a, b) + 1. (c, d)}   

_{= (4a, 4b) + (c, d)}

_{T(4, 1) =(4a+c, 4b+d)}