Question

T: R^2 -> R^2 be the linear transformation that is described by a counter clockwise rotation through pi divided by 2. The matrix of T with respect to the standard basis has 0 and 1 in its first column, and -1 and 0 in its second. Given an example, with explanation, of a basis with respect to which the matrix of T has 2 as its upper left entry.

Answer 1

To find a basis that results in a matrix with 2 as the upper left entry for the rotation transformation, multiply the original basis by √2 to obtain new basis vectors (2, 0) and (0, 2).

Step-by-step explanation:

The student is asking for an example of a basis with respect to which the matrix of a linear transformation T, representing a counter-clockwise rotation by π/2, has 2 as its top left entry. To change the matrix of a linear transformation, we need to change the basis. If the original basis is (1, 0) and (0, 1), we can multiply these basis vectors by a scalar to obtain a new basis. For example, if we want the top left entry to be 2, we can multiply the original basis vectors by √2. This would give us the new basis vectors (2, 0) and (0, 2).

In this case, the matrix of T with respect to this new basis would be:

[2, 0]
[-2, 0]

That's because the rotation would effectively swap the components and negate the second component of the vectors in the original basis. When the original basis vectors are scaled, this effect is also scaled, resulting in 2 as the upper left entry in the matrix.