36.1: Pre-class Review
- Page ID
- 70224
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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:
\[d(u,v) = \| u-v \| = \sqrt{\langle u-v, u-v \rangle} \nonumber\]