
A Geometric Review of Linear Algebra [pdf] - ivan_ah
http://www.cns.nyu.edu/~eero/NOTES/geomLinAlg.pdf
======
ivan_ah
The A in the title is for AWESOME.

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.

~~~
yxhuvud
All the essential topics? I miss eigenvalues (and probably determinants as a
stepping stone to that). Dunno if the most pedagogical approach would be
before or after SVD though.

~~~
icegreentea
At least from a non-software engineering context, you're usually taught
eigenvalues first as they are quite useful for a lot different problems...
basically all linear systems.

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.

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julie1
euuuhhhh, are people really not been taught this? Eigenvalues/Eigenvector,
space rotation, n-dimension dot product with missing dimensions, LU/RU system
...

How can people do computer science without this knowledge? It is needed for

1) solving equations;

2) transforming 3D vector and then projecting them on a 2D space (a screen) ;

3) text indexation;

4) resolving graph problem (like drawing a CPU based on declaring connection
between "wires"...

5) and with dual space you access even more fire power (fourier, laplace...);

6) clustering db is homeomorph to a balanced n-partition of a graph;

Geometry is one of the most powerful tool prior to the knowledge of any
programming language in computer industry.

~~~
judk
This link IS people being taught this. Don't be so dismissive of people whose
lives are slightly different from yours.

~~~
julie1
PS I do AM surprised this knowledge is not a requirement for CS.

Like knowing non linear problem (like estimating the cost of resource "per
user" on the cloud with all their "discrete" rough edged pricing plan) cannot
be solved with linear algebra.

The cloud, the cost of IP transit (95th percentile) are defeating by nature
any possible estimations. And still I see PhD and geniuses trying to code with
linear equations the pricing based on the evolution of users. This is
mathematically impossible.

So am I a genius, a dunce or what? I would be glad to know I am wrong, I would
stop wondering.

~~~
revelation
Of course non-linear problems can be solved in linear algebra. This is the
basis of the Gauss–Newton algorithm, which reduces a non-linear problem to the
repeated solving of a linear system.

~~~
julie1
estimation through linear techniques don't work on non linear problems. Else,
if I have X users, tell me the formula to guess my IP transit bill when the
number of users double?

If you have it, you are a genius.

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mikevm
Maybe this book will also be of interest: [http://www.amazon.com/Practical-
Linear-Algebra-Geometry-Tool...](http://www.amazon.com/Practical-Linear-
Algebra-Geometry-Toolbox/dp/1466579560)

~~~
ctrijueque
I'm using this book as a companion to this edX course:

[https://courses.edx.org/courses/UTAustinX/UT.5.01x/1T2014/in...](https://courses.edx.org/courses/UTAustinX/UT.5.01x/1T2014/info)

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j2kun
I personally don't like how he uses the term "system" in place of "mapping" or
"transformation," but they're some good notes

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quarterwave
1\. The section on matrices should start (not end) with the orthogonal length
preserving co-ordinate transformation. It's the easiest geometry/trig
connection, the elements can be worked out by hand, demonstrates unitary
property, and makes a nice segue into inverse. 2\. The section on linear
system response could make a mention of Fourier series (even Fourier
integral). 3\. It's still a mystery to me why linear algebra is not taught
using the Dirac notation.

