A function $f$ is linear if $f(a{\bf x} + b{\bf y}) = af({\bf x}) + bf({\bf y})$
Or equivalently $f$ is linear if
1.) $f(a{\bf x}) = af({\bf x})$ and
2.) $f({\bf x} + {\bf y}) =
f({\bf
x}) + f({\bf y})$
Theorem: If $f$ is linear, then $f({\bf 0}) = {\bf 0}$
Proof: $f({\bf 0}) = f(0 \cdot {\bf 0}) = 0 \cdot f({\bf 0}) = {\bf 0}$
Example 1.) $f :R \rightarrow R$, $f(x) = 2x$ is linear.
Proof: $f(ax + by) = 2(ax + by) = 2ax + 2by = af(x) + bf(y)$
Example 2.) $g :R \rightarrow R$, $g(x) = \sqrt{x} $ is NOT linear.
$g(x + y) = \sqrt{x + y} \not= \sqrt{x} + \sqrt{y}$
Proof: $g(1 + 4) = g(5) = \sqrt{5}$, but $g(1) + g(4) = \sqrt{1} + \sqrt{4} = 1 + 2 = 3$
Example 3.) $h :R \rightarrow R$, $h(x) = \frac{1}{x}$ is NOT linear.
Proof: $\frac{1}{3} = \frac{1}{1 + 2} \not= \frac{1}{1} + \frac{1}{2}$
Example 4.) $j :R \rightarrow R$, $j(x) = e^x$ is NOT linear.
Proof 1: $j(1+1) = j(2) = e^{2} \not= 2e^1 = e^1 + e^1 = j(1) + j(1)$
Proof 2: $j(2(1)) = j(2) = e^{2} \not= 2e^1 = 2j(1)$
Example 5.) $k :R \rightarrow R$, $k(x) = ln(x)$ is NOT linear.
Proof 1: $k(1+1) = k(2) = ln(2) \not= 0 = ln(1) + ln(1) = k(1) + k(1)$
Proof 2: $k(2(1)) = k(2) = ln(2) \not= 0 = 2ln(1) = 2k(1)$