Let L: V → W be a linear map. 

(a) Suppose V,W are finite-dimensional vector spaces, and let A be a matrix representative of L. Explain why we can identify coker A ≃ W/img A and coimg A = V/ker A as quotient vector spaces, cf. 

These characterizations are used to give intrinsic definitions of the cokernel and coimage of a general linear function L: V → W without any reference to a transpose (or, as defined below, adjoint) operation. Namely, set cokerL ≃ W/img L and coimg L = V/ kerL.

(b) The index of the linear map is defined as indexL = dimkerL − dimcokerL, using the above intrinsic definitions. Prove that, when V,W are finite-dimensional, indexL = dimV − dimW.