Final answer:
To determine the matrix representing a linear operator L in R², apply L to the basis vectors and structure the outputs as columns of the matrix A. The process varies for different bases, and finding a matrix for the null space basis would result in the zero matrix unless L is trivial.
Step-by-step explanation:
To find the matrix A representing the linear operator L with respect to different bases in R², we need to apply the linear operator to each of the basis vectors and express the results as columns of the matrix in the specified basis.
Standard Basis
For the standard basis (e1 = [1, 0], e2 = [0, 1]) of R², we apply L to each basis vector and the outputs L(e1) and L(e2) will be the columns of the matrix A.
Arbitrary Basis
For an arbitrary basis B = {b1, b2}, we would first apply L to b1 and b2, then express these outputs in terms of the basis B, forming the matrix A with respect to B.
Non-Square Matrix
A non-square matrix does not apply to an operator on R² since we expect a 2x2 matrix; however, if we're considering a transformation from R² to another space, we may have a non-square matrix.
Null Space Basis
The basis for the null space consists of vectors that are mapped to the zero vector by L. If L is nontrivial, its null space may be trivial (just the zero vector), in which case this part of the question is not applicable. Otherwise, the matrix A with respect to the null space basis would be the zero matrix.