Question

Consider the multiplication operator

LA : R² → R² defined by LA(x) = Ax where
A= [ -1 -2 ]. Find an ordered basis
[ -3 3 ]
B = (b1, b2) for R² such that
[LA]B = [ -18 -13 ] Hint: [LA]E=A
[ 27 20 ]
where E is the standard basis.
b1 = [ 1 ], b2 = [ -11 ]
[ 10 ] [ __ ]

Answer 1

The question involves finding an ordered basis for R² that results in a particular matrix representation of a linear transformation. The standard transformation matrix is known, but solving for the basis requires matrix operations and change of basis, which is not explicitly detailed in the question.

Step-by-step explanation:

The student is asking for assistance in finding an ordered basis B = (b1, b2) for R² such that the matrix representing the linear transformation L_A with respect to this basis is a specific 2x2 matrix. By utilizing the hint provided, that the standard matrix for the transformation is given by A, we can infer that the matrix representation of L_A with respect to basis B should be similar to A and have the same eigenvalues.

Given that [L_A]_B is provided, and assuming b1 and b2 are already known, the task becomes to find the entries of b2 such that when the transformation is applied to the new basis vectors, the new matrix representation is achieved. For this, several mathematical operations involving change of basis and matrix multiplication would be needed to solve for the unknown entries of the basis vectors.

The direct computation, however, is not provided in the question, so final steps that include expressing each standard basis vector as a linear combination of the new basis vectors and applying the transformation matrix to these new basis vectors in order to derive the required matrix representation cannot be conducted without this additional information.