36.1: Pre-class Review
- Page ID
- 70224
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)An inner product on a real vector space \(V\) is a function that associates a number, denoted as \(\langle u,v \rangle\), with each pair of vectors \(u\) and \(v\) of \(V\). This function satisfies the following conditions for vectors \(u,v,w\) and scalar \(c\):
\[\langle u,v \rangle = \langle v,u \rangle \text{ symmetry axiom} \nonumber\]
\[\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle \text{ additive axiom} \nonumber\]
\[\langle cu,v \rangle = c\langle v,u \rangle \text{ homogeneity axiom} \nonumber\]
\[\langle u,v \rangle = \langle v,u \rangle \text{ Symmetry axiom} \nonumber\]
\[\langle u,u \rangle \ge 0 \text{ and } \langle u,u \rangle = 0 \text{ if and only if } u = 0 \text{ positive definite axiom} \nonumber\]
The dot product of \(R^n\) is an inner product. However, we can define many other inner products.
Norm of a vector
Definition: Let \(V\) be an inner product space. The norm of a vector \(v\) is denoted by \(\| v \|\) and is defined by:
\[\| v \| = \sqrt{\langle v,v \rangle}. \nonumber\]
Angle between two vectors
Definition: Let \(V\) be a real inner product space. The angle \(\theta\) between two nonzero vectors \(u\) and \(v\) in \(V\) is given by:
\[cos(\theta) = \frac{\langle u,v \rangle}{\| u \| \| v \|}. \nonumber\]
Orthogonal vectors
Definition: Let \(V\) be an inner product space. Two vectors \(u\) and \(v\) in \(V\) are orthogonal if their inner product is zero:
\[\langle u,v \rangle = 0. \nonumber\]
Distance
Definition: Let \(V\) be an inner product space. The distance between two vectors (points) \(u\) and \(v\) in \(V\) is denoted by \(d(u,v)\) and is defined by:
\[d(u,v) = \| u-v \| = \sqrt{\langle u-v, u-v \rangle} \nonumber\]