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S2.1

math.toronto.edu · 4,889 words · saved by 1 readers

First, a remark. In high school, a function ℓ:ℝ→ℝ ℓ:R→R is called linear if it has the form ℓ(x)=mx+b ℓ(x)=mx+b . One could make the same definition for functions ℝ n → ℝ m Rn→Rm ; then a function ℓ: ℝ n → ℝ m ℓ:Rn→Rm is linear if it has the form ℓ(x)=Mx+b ℓ(x)=Mx+b wbere M M is a m×n m×n matrix and b∈ ℝ m b∈Rm .

S2.1 2.1: Differentiation $\newcommand{\R}{\mathbb R }$ $\newcommand{\N}{\mathbb N }$ $\newcommand{\Z}{\mathbb Z }$ $\newcommand{\bfa}{\mathbf a}$ $\newcommand{\bfb}{\mathbf b}$ $\newcommand{\bff}{\mathbf f}$ $\newcommand{\bfu}{\mathbf u}$ $\newcommand{\bfx}{\mathbf x}$ $\newcommand{\bfy}{\mathbf y}$ $\newcommand{\bfm}{\mathbf m}$ $\newcommand{\bfh}{\mathbf h}$ $\newcommand{\ep}{\varepsilon}$ Differentiation of real-valued functions Differentiability and the gradient Partial derivatives Differentiability vs. partial differentiability Directional derivatives, and the meaning of the gradient Pro

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