Final answer:
To determine compatibility conditions for a system of linear equations, the RREF of the matrix is analyzed to study the rank and null space, guided by the Rank-Nullity Theorem.
Step-by-step explanation:
To determine compatibility conditions for a system of linear equations by computing a basis for the cokernel of the coefficient matrix, you would typically look at the RREF (Row-Reduced Echelon Form) of the matrix. This allows you to examine the rank of the matrix and its null space. The Rank-Nullity Theorem is relevant here because it relates the rank of a matrix, which is the dimension of its row space (and also its column space), to the nullity, which is the dimension of the null space of the matrix. The basis conditions are that the number of free variables corresponds to the dimension of the cokernel, and this is given by the difference between the number of columns and the rank of the matrix.
In summary, to establish a basis for a cokernel, you don't look at eigenvalues or eigenvectors, which are related to the different question of diagonalization and the intrinsic properties of linear transformations. Instead, you focus on the Rank-Nullity Theorem and the RREF of the matrix to reveal the row space and null space, which together determine the compatibility of the system.