I have been writing about linear algebra almost non-stop for the past two months to organize the best possible "playlist" of topics that cover a linear algebra course. It's a hard problem, this linearization of a graph of dependencies into a stream of .tex
Prof. Simoncelli has a very good geometrical take on the problem. He manages to cover all the essential topics in 10 pages. Geometrically speaking, vectors = arrows and matrices = transformations on arrows.
They're not missing just judiciously omitted. Actually seeing SVD presented in an introduction is what inspired me to post this.
The SVD approach works for an matrix A in R^{m x n}
A = UΣV^T
A = UΛU^T
At least the way our curriculum was arranged, in first year had a 'simple' linear algebra course that consisted of solving systems of linear equations (like Gaussian Elimination), and 'simple' things with matrices and vectors, which then ended in this rush to teach eigen-decomposition, which we pretty much just learned by rote. I don't believe anyone really had a feel for what we were actually doing.
But the flip side was that we learned eigen-decomposition in time for all the 'fun' applications of it, and by the time our far more rigorous linear algebra course occurred in 3rd year (which ended with SVD), when we went over eigen-decomp again, it was like a brilliant 'OH, I GET ITT' moment for everyone in the class.
{arithmetic}
|
{vecops} = {Vector operations}
{dotproduct}
{arithmetic}
|
{RREF} = {Reduced row echelon form}
{arithmetic}
| {patience}
| | {dotproduct}
| | |
{matrices}
{Mops} = {Matrix operations}
{Meqns} = {Matrix equations}
{Mmult} = {Matrix multiplication}
{Mdet} = {Determinants}
{Minv} = {Matrix inverse}
{y=mx+b}
|
{linear} = {Linearity}
{plane} = {Lines and planes}
{dotproduct}
|
{proj} = {Projections}
{basis} = {Coordinate projections}
{vectors}
|
{VS1} = {Vector spaces}
{VS2} = {Abstract vector spaces} // knowledge buzz moment
|
| {dotproduct}
| |
{IPS} = {Abstract inner product spaces}
{y=mx}
| {Mmult}
| | {function f:R->R}
| | |
{linT} = {Linear transformations}
{VSRREF} = {Vector space techniques for matrices}
{TasM} = {Finding matrix representations}
{basis2} = {Change of basis for matrices}
{Mmult}
| {linT}
| |
{eigV} = {Eigenvalues and eigenvectors}
{Mdecomp}= {Matrix decompositions}
|
{specM} = {Special types of matrices}
{VS1}
|{plane}
||{eigV}
|||{linT}
|||||{Meqns}
||||||{Mmult}
||||||||{Mdet}
|||||||||
{invMthm} = {Invertible matrix theorem}
{proj}
|
{GSorth} = {Gram--Schmidt orthogonalization}
{RREF}
|
{LP} = {Linear programming}
{Mmult}
|{Meqn}
||{RREF}
|||
{solve} = {Solving systems of equations}
{LSaprx} = {Least squares approximate solution}
{ML} = {Machine learning}
{LAoverZ_q}= {Linear algebra over a finite field}
|||
||{ECCs} = {Error correcting codes}
|{crypto} = {Cryptography}
{netcode}= {Network coding}
{trig}
|{taylor}
||{AVS}
|||{IPS}
|||||
{fourier}= {Fourier series}
{fourier}
| {prob} = {Probability density}
| | {LAoverC}= {Linear algebra with complex numbers}
| | |
{quantum}= {Quantum mechanics}