Question

Suppose T is a linear transformation with T(1,0) = (2,3) and T(0,1) = (4,5). Find the standard matrix [T] such that T(x,y) = [T][x,y].

Answer 1

The standard matrix [T] for the linear transformation T given T(1,0) = (2,3) and T(0,1) = (4,5) is a 2x2 matrix whose columns are the images of the standard basis vectors: [T] = |2 4| |3 5|.

Step-by-step explanation:

To find the standard matrix [T] for a linear transformation T that maps from 2D vectors to 2D vectors, you can use the images of the standard basis vectors under the transformation. Given that T(1,0) = (2,3) and T(0,1) = (4,5), we can infer that the first column of [T] is (2,3) and the second column is (4,5), as these are the images of the standard basis vectors (1,0) and (0,1), respectively.

The standard matrix [T] can be represented as:

[T] = |2 4|
|3 5|

This matrix allows us to compute the image of any vector (x,y) in R² under the transformation T by multiplying [T] with the coordinate column vector [x,y].