---
title: "Axler 6.B"
source: https://www.jemoka.com/posts/kbhaxler_6_b/
---

OMG its Gram-Schmidtting
Key Sequence we defined lists of vectors that all have norm 1 and are all orthogonal to each other as orthonormal; we showed orthonormal list is linearly independent by hijacking pythagoras of course, once we have a finitely long linearly independent thing we must be able to build a basis. The nice thing about such an orthonormal basis is that for every vector we know precisely what its coefficients have to be! Specifically, \(a_{j} = \langle v, e_{j} \rangle\). That&rsquo;s cool. What we really want, though, is to be able to get an orthonormal basis from a regular basis, which we can do via Gram-Schmidt. In fact, this gives us some useful correlaries regarding the existance of orthonormal basis (just Gram-Schmidt a normal one), or extending a orthonormal list to a basis, etc. There are also important implications (still along the veins of &ldquo;just Gram-Schmidt it!&rdquo;) for upper-traingular matricies as well We also learned, as a result of orthonormal basis, any finite-dimensional linear functional (Linear Maps to scalars) can be represented as an inner product via the Riesz Representation Theorem, which is honestly kinda epic. New Definitions orthonormal + orthonormal basis Gram-Schmidt (i.e. orthonormalization) linear functional and Riesz Representation Theorem Results and Their Proofs Norm of an Orthogonal Linear Combination An orthonormal list is linearly independent An orthonormal list of the right length is a basis Writing a vector as a linear combination of orthonormal basis Corollaries of Gram-Schmidt Every Inner Product Space has an orthonormal basis Orthonormal list extended to orthonormal basis Orthonormal upper-triangular matrix basis exists if normal upper-triangular exists Schur&rsquo;s Theorem Riesz Representation Theorem Questions for Jana Interesting Factoids
