For this general solution for x
and y to be valid,
the denominator ad − cb
apparently has to be different from zero.
5.1.2 Two–by–Two Matrices: Definition
We write the two unknowns x
and y as the components of
a two–dimensional vector x,
\begin{eqnarray}
x := \left (\array{
x\cr
y } \right ).& & %&(5.4)\\
\end{eqnarray}
Then, we write the two constants e
and f as the components of
a two–dimensional vector v
\begin{eqnarray}
v := \left (\array{
e\cr
f } \right ).& & %&(5.5)\\
\end{eqnarray}
The two–by–two system of linear equations, Eq. (5.1), maps the vectorxonto thevector v.
We write this in the following abstract form:
\begin{eqnarray}
Ax& =& v ⇔\left (\array{
a&b\cr
c &d } \right )\left (\array{
x\cr
y } \right ) = \left (\array{
e\cr
f } \right ),%&(5.6)\\
\end{eqnarray}
where we defined the two–by–two matrix
\begin{eqnarray}
A := \left (\array{
a&b\cr
c &d } \right ).& & %&(5.7)\\
\end{eqnarray}
A two–by–two matrix is a quadratic scheme which, upon operating on a vector
x on its right, transforms this
vector into another vector v
according to the rule
By comparison we recognise that this matrix equation,
Ax = v, is
equivalent to the system Eq.(5.1).
5.1.3 Linear Mappings and Matrix Operatings
A linear mapping A from {ℝ}^{2} → {ℝ}^{2}
maps a vector x onto the vector
Ax. The mapping is represented
by a two-by-two matrix A.
The mapping must fulfill {x}_{1} +{ x}_{2} → A({x}_{1} +{ x}_{2}) = A{x}_{1} + A{x}_{2},\quad λx → A(λx) = λAx,\quad λ ∈ ℝ.
The above can be generalised (trivially) to complex matrices; the mapping is then from
{ℂ}^{2} → {ℂ}^{2}, and
λ can also
be complex