Chapter 1 - Linear Equations in Linear Algebra - 1.8 Exercises - Page 70: 29

See solution

a) $f(c x+d y)=m(c x+d y)=m c x+m d y$ a) When $b=0, f(x)=m x .$ In this case, for all $x$ $y$ in $R$ and all scalars $c$ and $d$ \[ \begin{array}{l} =c(m x)+d(m y) \\ =c \cdot f(x)+d \cdot f(y) \end{array} \] This shows that $f$ is linear b) When $f(x)=m x+b$ with b nonzero, $f(0)=$ \[ m(0)+b=b \neq 0 \] c) In calculus, $f$ is called a "linear function" because the graph of $f$ is a line.