adjoint identities
The adjoint is not just a definition; it obeys a tidy algebra worth memorizing. It is conjugate-linear in scalars, (c T)^* = conj(c) T^*; additive, (S + T)^* = S^* + T^*; order-reversing on products, (S T)^* = T^* S^*; involutive, (T^*)^* = T; and it sends identity to identity. These mirror the transpose rules with conjugates sprinkled in.
The order reversal (S T)^* = T^* S^* is the one people slip on. It is the same flip as for the inverse or transpose: apply S then T, and the adjoint unwinds them in reverse. A quick check using <S T u, v> = <u, (ST)^* v> confirms the swap.
The deepest identities are the four fundamental subspace relations: ker T^* = (im T)-perp and im T^* = (ker T)-perp, with the two companions ker T = (im T^*)-perp and im T = (ker T^*)-perp. They say the adjoint exactly trades kernels for image-complements.
These relations are the orthogonal upgrade of Vol I's four fundamental subspaces. They make the rank-nullity bookkeeping geometric: the domain splits as ker T (+) im T^*, the codomain as im T (+) ker T^*, and T is a clean isomorphism between the two non-kernel pieces. Least squares, the SVD, and the pseudoinverse all rest on exactly this.
The algebra of adjoints plus the fundamental-subspace duality; together they make rank-nullity into an orthogonal decomposition.
Memory hook: ker T^* = (im T)-perp says the adjoint's null directions are exactly those the original operator never reaches. This single relation is why least squares lands the residual orthogonal to the column space.