Let T be a linear operator on R2 defined as follows on the standard basis of R2. T(1, 0) = (3, 2), T(0, 1) = (−1, 4). Find T(9, 4).

(9,4) = 9*(1,0) + 4*(0,1)

So,
T(9,4) = 9T(1,0)+4T(0,1)
= 9(3,2) + 4(-1,4)
= (27,18) + (-4,16)
= (23,34)

To find T(9, 4), we need to apply the linear operator T to the vector (9, 4).

First, we need to determine how T acts on the standard basis vectors (1, 0) and (0, 1).

Given that T(1, 0) = (3, 2) and T(0, 1) = (-1, 4), we can use these values to express any vector in R2.

To find T(9, 4), we can express the vector (9, 4) in terms of the standard basis vectors.

(9, 4) = 9(1, 0) + 4(0, 1)

Then, we can use linearity to apply the linear operator T to each term of the sum:

T(9, 4) = 9 * T(1, 0) + 4 * T(0, 1)

Substituting the values we know:

T(9, 4) = 9 * (3, 2) + 4 * (-1, 4)

Now, we can perform the scalar multiplication and addition:

T(9, 4) = (27, 18) + (-4, 16)

Adding the corresponding components:

T(9, 4) = (27 + (-4), 18 + 16)

T(9, 4) = (23, 34)

Therefore, T(9, 4) = (23, 34).