“Guassian elimination” typo
This is the 2-week intro to linear algebra I received in Grade 12. The real interesting stuff starts with those eigen-things, which leads you to solving interesting problems in time series analysis and systems of ecology, among others.
… actually it’s Gauss-Jordon Elimination. Gaussian elimination would only give you an Upper Trianglular Matrix instead of an Identity Matrix.
I Love Linear Algebra but until the K-12 system gets a clue have taken to promoting column vectors as often as possible. Points become x stacked on y stacked on z and of course one can always do the transpose it the medium makes row vectors more palatable. And lets not forget to pay homage to Gilbert Strang in these discussion as one who didn’t need to but stuck his folksy lecture out at MIT Open Courses.
“Linear algebra emerged in the 1800s yet spreadsheets were invented in the 1980s. I blame the gap on poor linear algebra education.”
Spreadsheets have been used by accountants for hundreds of years ( http://dssresources.com/history/sshistory.html ), and programs for computers were developed almost as soon as there were computers.
“However, linear algebra is mainly about matrix transformations, not solving large sets of equations (It’d be like using Excel for your shopping list).”
I’ve used it to solve large sets of equations, with thousands of equations, but I’ve also used Excel for a shopping list too… 
Long-lost reply (I lost my laptop the day after I posted this article…argh).
@Frederick: Thanks for the note, and the detailed examples! There’s definitely lots to explore – I’m barely getting my toes wet – and I like the analogy of a “cone” of possibilities. Also, the idea that a curved space is not a vector space, but its tangent space is – pretty cool transformation. So much of math is just shifting your perspective.
@Bill: Those eigen-things seem to be the heart of it all.
@D.Dick: Thanks, fixed.
@Jeremy: Matrixes can definitely go deeper (to any linear operation) but it’s a crawl/walk/run thing.
@Alex: Thanks so much! Appreciate the detailed overview. I’ll have to dive into it.
@Sriram: Glad it clicked, and thanks for the link.
@SDX2000: Really appreciate it :).
@Mentock: Good point. Maybe a better phrasing is that spreadsheets have been used by accountants for centuries, without them realizing they could have been helped by “linear algebra” :).
@unconed: No problem. 99% of linear algebra courses will use vectors / projections, but I like spreadsheets because they’re so tangible and familiar. We should use every analogy we can.
@Ann: Thanks for the report, I’ll take a look.
@Ilya: Welcome!
@brian m: Yep, matrixes started off as bookkeeping for equations. And math is definitely a tool/language for communication. If we’re using math, but missing the ideas, we’re not doing math!
@Neo: Thanks, fixed.
@mark: Thanks for the reminder, I need to revisit the Strang lectures :).
@Tom, @George: Yep, “Grammar” was my analogy for focusing on structure but not ideas. Maybe I can think about the transition there.
Thanks for sharing your insights on matrices…reading this brought a tear to my eye…I wish my school teachers were like you.
Beautifully explained, as usual. Thank you.
A good accompaniment to your explanation is a geometric intuition of matrices, eigenstuff and singular value decomposition here: http://www.ams.org/samplings/feature-column/fcarc-svd
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Thanks for this article. You have a typo in this sentence:
A determinant of 0 means matrix is “desctructive” and cannot be reversed (similar to multiplying by zero: information was lost).
desctructive => destructive
OMG in a matter of just a few lines you’ve completely de-mystified the notion of eigenvector. Thanks!
Cool articles. Could you do one that covers isomorphism, monomorphism epimorphism and so on?
Hi Abdul, great comment. Yes, a reflection is a good example – we stay along the same line, but are pointing the other way. Appreciate the clarification.
Hi, and thank you for making this article.
I’d like to point out that an eigenvector is a vector whose direction is unchanged or invariant under a transformation.
An invariant line is one where any point on the line is mapped to another point on the same line. This means that under a transformation, a vector could change its direction to point in the opposite direction (and this would also mean it would be on the same line), and hence this vector would also be an eigenvector and have a corresponding eigenvalue which would be negative in this case (the vector would be scaled in the opposite direction).
I thought I should mention this as your explanation (and the wiki demo) is quite misleading as it only demonstrates one of the two possible cases (direction being the same)
Thanks
This is epic! Simply epic! Linear Algebra I got you now…
This has literally blown my mind multiple times, on multiple levels. This is exactly what I’ve needed… so glad I found this before the final exam lol. Thank you so much for posting this, keep up the good work!
This is not giving the correct intuition for linear algebra.
See gilbert strang’s first couple of lectures. They give an intuitive feel and are presented by someone who really understands linear algebra.
Hi Sam! I think intuition clicks different – if one analogy helps elucidate an aspect of the subject, so much the better (it’s not like you’re limited to one metaphor). I like Strang’s work in general, but didn’t have much intuition even after acing my university class that used his book! There’s more metaphors I need to find for myself.
Looking at matrices as “operations” that take “input” data and transform to “output” data, is very intuitive.