Chapter 6 Spring

wait, we just got the definition of the dot product? in chapter 6??

we can generalize the dot product to get the inner product

the inner product is more fundemental than lenght, and can in fact lead to the concepts of len and angles

we denote this inner product with \(\langle u, v \rangle\). now we get to the definition of the inner product:

takes an inner product of two elements $\in V$ and goes to a number $\langle u,v \rangle \in F$
has the following properties:
**positivity**                      $\langle v, v \rangle \geq 0$ for all $v \in V$

**definiteness**                    $\langle v, v \rangle = 0$ iff $v = 0$

**additivity in first slot**        $\langle u+v, w \rangle = \langle u, w \rangle + \langle v, w \rangle$ for all $u, v, w \in V$

**homogeneity in the first slot**   $\langle \lambda v, u \rangle = \lambda \langle v, u \rangle$ for all $\lambda \in F$ and all $u, v \in V$

**conjugate symmetry**              $\langle u, v \rangle = \overline{\langle v, u \rangle}$ for all $u, v \in V$

since all real numbers equal their complex conjugate, we can just say that in real vector spaces \(\langle u, v \rangle = \langle v, u \rangle\)

now with the inner product, we can define an inner product space which is just a

vector space along with an inner product

V is an inner product space for the rest of the chapter

the func that takes \(v\) to \(\langle v, u \rangle\) is a linear map from \(V\) to \(F\)

each inner product also determines a norm, following the pattern \(||v|| = \sqrt{\langle v, v \rangle}\)

the norm is also also ~homogenous: \(||\lambda v || = |\lambda| \text{ }||v||\)

we also get to define the concept of orthogonality:

title: orthogonal
two vecs are othogonal if $\langle u, v \rangle = 0$

ending on 169 *

iteresting, 0 is the only vec orthogonal to itself

with orthogonal decomposition, which i dont really get yet, we can prove the cauchy-schwarz inequality!

title: cauchy-scharz inequality
$$|\langle u,v \rangle| \leq ||u|| \space ||v||$$
holds iff u or v is a scalar multiple of the other.

and now we get to prove all kinds of geometric properties, like the triangle equality, the parelleogram equailty, ect