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’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 “just Gram-Schmidt it!”) 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
- Riesz Representation Theorem