  In this paper we derive some basic results of circuit theory using `Implicit
Linear Algebra' (ILA). This approach has the advantage of simplicity and
generality. Implicit linear algebra is outlined in [1]. We denote the space of
all vectors on $S$ by $\mathcal{F}_S$ and the space containing only the zero
vector on $S$ by $\mathbf{0}_S.$ The dual $\mathcal{V}_S^{\perp}$ of a vector
space $\mathcal{V}_S$ is the collection of all vectors whose dot product with
vectors in $\mathcal{V}_S$ is zero. The basic operation of ILA is a linking
operation ('matched composition`) between vector spaces
$\mathcal{V}_{SP},\mathcal{V}_{PQ}$ (regarded as collections of row vectors on
column sets $S\cup P, P\cup Q,$ respectively with $S,P,Q$ disjoint) defined by
$\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ}\equiv
\{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (g_P,h_Q) \in \mathcal{V}_{PQ}\},$
and another ('skewed composition`) defined by
$\mathcal{V}_{SP}\rightleftharpoons \mathcal{V}_{PQ}\equiv
\{(f_S,h_Q):((f_S,g_P)\in \mathcal{V}_{SP}, (-g_P,h_Q) \in \mathcal{V}_{PQ}\}.$
The basic results of ILA are the Implicit Inversion Theorem (which states that
$\mathcal{V}_{SP}\leftrightarrow(\mathcal{V}_{SP}\leftrightarrow
\mathcal{V}_S)= \mathcal{V}_S,$ iff $\mathcal{V}_{SP}\leftrightarrow
\mathbf{0}_P\subseteq \mathcal{V}_S\subseteq
\mathcal{V}_{SP}\leftrightarrow\mathcal{F}_S$) and Implicit Duality Theorem
(which states that $(\mathcal{V}_{SP}\leftrightarrow \mathcal{V}_{PQ})^{\perp}=
(\mathcal{V}_{SP}^{\perp}\rightleftharpoons \mathcal{V}_{PQ}^{\perp}$). We show
that the operations and results of ILA are useful in understanding basic
circuit theory. We illustrate this by using ILA to present a generalization of
Thevenin-Norton theorem where we compute multiport behaviour using adjoint
multiport termination through a gyrator and a very general version of maximum
power transfer theorem, which states that the port conditions that appear,
during adjoint multiport termination through an ideal transformer, correspond
to maximum power transfer.
