Linear Functions¶
Recall (doc/postdoc). Not a first course.
Job: affine maps \(f(x)=mx+b\) on \(\mathbb{R}\); “linear” in school = affine; through origin \(\Leftrightarrow\) \(K\)-linear.
Core¶
| Form | Equation |
|---|---|
| Slope-intercept | \(y=mx+b\) |
| Point-slope | \(y-y_1=m(x-x_1)\) |
| Two-point | \(m=\dfrac{y_2-y_1}{x_2-x_1}\) (\(x_1\neq x_2\)) |
| Standard | \(Ax+By=C\) |
| Vertical | \(x=a\) (not a function \(y=f(x)\); infinite slope) |
Slope \(m=\tan\theta\) for inclination \(\theta\); parallel \(\Leftrightarrow\) same \(m\); perpendicular (Euclidean) \(m_1m_2=-1\) (neither vertical).
As maps: \(f(x)=mx+b\) is \(K\)-linear iff \(b=0\). Affine group: \(x\mapsto mx+b\), \(m\neq0\).
Used later for¶
- Systems / matrices: \(Ax+b\)
- Calculus: derivative as best linear approximation
- Geometry: lines in \(\mathbb{R}^2\)
- Functional analysis: bounded linear operators (through origin)
Check¶
Why is \(f(x)=2x+3\) not linear as a map of vector spaces?
\(f(0)=3\neq0\) and \(f(x+y)\neq f(x)+f(y)\). It is affine: \(f(x)=L(x)+f(0)\) with \(L(x)=2x\) linear.