Find a linear transformation 7: R2 R2 such that 7(1,-2) = (1, 1) and T(0, 1) = (2,3), where B = {(1,-2), (0, 1)) is a basis of IR2.

To find a linear transformation T: R2 -> R2 such that T(1,-2) = (1, 1) and T(0, 1) = (2,3), we can use the fact that T is completely determined by its action on a basis of R2. In this case, the basis B = {(1,-2), (0, 1)} is given.

Step 1: Write the vectors T(1,-2) and T(0, 1) as column vectors:

T(1,-2) = (1, 1) becomes [1, 1]^T

T(0, 1) = (2,3) becomes [2, 3]^T

Step 2: Form the matrix A of the linear transformation T by putting these column vectors side by side:

A = [1 2] [1 3]

This is the matrix of the linear transformation T with respect to the basis B.

Step 3: To find the standard matrix of the linear transformation T, we need to change the basis from B to the standard basis E = {(1,0), (0,1)}. This can be done by a change of basis matrix P, which is formed by writing the vectors of B as column vectors:

P = [1 0] [-2 1]

Step 4: The standard matrix of T is then given by the product P^-1 * A * P, where P^-1 is the inverse of P.

So, the first step is to find the inverse of P:

P^-1 = 1/(1*(-1) - 0*(-2)) * [1 -0] [2 1] = [1 0] [2 1]

Step 5: Now, compute the product P^-1 * A * P:

P^-1 * A * P = [1 0] * [1 2] * [1 0] [2 1] [1 3] [-2 1] = [1 2] [3 5]

So, the standard matrix of the linear transformation T is [1 2; 3 5]. This means that T(x, y) = (x + 2y, 3x + 5y) for any (x, y) in R2.