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Section 8.7 Linear Operators (Advanced)

Origin of the Word “Linear” in Linear Operator.

The word linear comes from linear equations, i.e. equations for straight lines. The equation for a line through the origin \(y=mx\) comes from the operator \(f(x)=mx\) acting on vectors which are real numbers \(x\) and constants that are real numbers \(\alpha\text{.}\) The first property:

\begin{equation} f(\alpha x)=m(\alpha x)=\alpha (mx)=\alpha f(x)\tag{8.7.1} \end{equation}

is just commutativity of the real numbers. The second property:

\begin{equation} f(x_1+x_2)=m(x_1+x_2)=m(x_1)+m(x_2)=f(x_1)+f(x_2)\tag{8.7.2} \end{equation}

is just distributivity of multiplication over addition. These are properties of real numbers that you've used since grade school, even if you didn't know to call the properties by these fancy names! Note that the equation for a line NOT through the origin \(y=mx+b\text{,}\) leading to the operator \(g(x)=mx+b\text{,}\) is NOT linear.

\begin{equation} g(x_1+x_2)=m(x_1+x_2)+b \ne (mx_1+b)+(mx_2+b)=g(x_1)+g(x_2)\text{.}\tag{8.7.3} \end{equation}

Inhomogeneous Linear Differential Equations.

It will be helpful to remember the example of linear equations with zero and non-zero \(y\)-intercept when you are learning the difference between homogeneous and inhomogeneous linear differential equations in Section 9.1.

As with the example above for straight lines through the origin and not through the origin, if you take a homogeneous linear differential operator \(\LL\) as in example 1 above and add to it an inhomogeneous term \(b(x)\text{,}\) the resulting operator which takes \(y\) to \(\LL y - b(x)\) is NOT linear. If \(y_p\) and \(y_q\) are both solutions of

\begin{equation} \LL y=b(x)\text{,}\tag{8.7.4} \end{equation}

then

\begin{equation} \LL \left(y_p+y_q\right)=2b(x)\ne b(x)\text{.}\tag{8.7.5} \end{equation}

You can NOT add two solutions of an inhomogeneous differential equation and get another solution. However, you CAN add ANY solution of the homogeneous equation to a solution of the inhomogeneous equation to get another solution of the inhomogeneous equation.