2. Whenever we want to isolate the direction of something, we use a unit vector.

3. While it is useful to think of linearly independent sets as “small” and spanning sets as “large”, be careful not to try to apply the converse. {0} is very small, but linearly dependent. {v, 2v, 3v, 4v, 5v, …} is very large, but spans only a dimension 1 space.

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Coordinate vectors and sweets
A restaurant serves 2 desserts, cannoli and cookies. Bill and Bob walk in. Bill says, “I’ll take a cannoli and two cookies.” Bob says, “I’ll take 3 cannoli, no cookies for me.” Finally, Xavier walks into the restaurant and says, “Hmm, I’ll take 3 of what Bill’s having and 2 of what Bob’s having.” To figure how much to give Xavier, the server thinks “He needs 3*1 + 2*3 cannoli and 3*2 + 2*0 cookies.”

[of course the server really thinks “what is wrong with this guy?” but we’re in math fantasy land, so suspend your disbelief.]

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• all linear transformations take 0 to 0
• images of subspaces are subspaces (special example: image)
• preimages of subspaces are subspaces (special example: kernel)
• closing a set of vectors before or after applying a transformation gives the same vector space
• we may unambiguously determine the entirety of a linear transformation from its action on a basis (linear extension) — which is what makes matrix representation possible!
• having an inverse is equivalent to being bijective
• the rank of a matrix is definable by its rows as well as its columns
• inverses of isometries are also isometries (also uses bilinearity of inner product)

The post Applying Linearity appeared first on rweber.net.

	diagonalizable	not diagonalizable
invertible	I_n,	rotation about non-axis line by, say, 60 degrees in R^3, rotation 90 degrees in R^2:
not invertible	zero matrix,	rotation 90 degrees plus projection onto yz-plane in R^3:

The post Table of matrix possibilities appeared first on rweber.net.

These are both incorporated to some degree into the math club linear algebra talk I gave, but are much shorter and self-contained.

• Markov chains (text generation and Google PageRank)
• computer graphics (vector graphics creation and manipulation)

A friend gave me an apocryphal quote for the reason to shift to raster graphics: “I have 8 pixels. How do I make it look like a cat?”

The post Linear Algebra Presentations appeared first on rweber.net.

Suppose M is invertible. Then there is some M^-1 such that MM^-1 = I, the identity matrix. Then 1 = det(I) = det(MM^-1) = det(M)det(M^-1), and 1 cannot be obtained as a product of 0 with anything, so both M and M^-1 have nonzero determinant.

Now suppose M is (square and) noninvertible. Then the kernel of the transformation T that M represents includes some nonzero vector X, and we may build a basis B for our vector space such that B includes X. The matrix M’ for T relative to B includes a column of all 0s, corresponding to the position of X in the ordering of B, so det(M’) = 0. For appropriate change of basis matrices P, P^-1 we have M = PM’P^-1, so det(M) = det(PM’P^-1) = det(P)det(M’)det(P^-1) = 0.

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A) U = (R x R, +_U, *_U)
where (a₁, a₂) +_U (b₁, b₂) = (a₁ + 2b₁, a₂ + 3b₂)
and *_U is the usual scalar multiplication

B) V = (R x R, +_V, *_V)
where (a₁, a₂) +_V (b₁, b₂) = (a₁ + b₁ + 1, a₂ + b₂)
and *_V is the usual scalar multiplication

C) W = (R x R, +_W, *_W)
where (a₁, a₂) +_W (b₁, b₂) = (a₁ + b₁ + 1, a₂ + b₂ + 1)
and r*_W(a₁, a₂) = (ra₁ + r – 1, ra₂ + r – 1)

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Exercise: show that if a vector space (over the real numbers) has at least two vectors, it has infinitely many.

Simple proof: If V has two vectors, one of them is nonzero. The span of such a vector is infinite because there are infinitely many real numbers, and its span is contained in V, so V is infinite.

Sharp-eyed student question: how do we know that aX and bX can’t be the same, for a, b distinct scalars and X a nonzero vector?

After a certain amount of trying to produce a counterexample, a quest I knew was doomed to fail but thought might be illuminating, I produced a two step proof of the infinitude of the span.

Lemma: If for some vector X there is a nonzero real number a such that aX = 0, then for any real number b, bX = 0.

Proof: Suppose X, a are as in the lemma. Since a is nonzero we may divide by it. Then for any scalar b, bX = ((b/a)a)X = (b/a)(aX) = (b/a)0 = 0.

Claim: If X is a nonzero vector and a, b are distinct real numbers, then aX, bX are distinct.

Proof: By the lemma, since 1X is not 0 (this is an axiom of vector spaces), no nonzero real number can give 0 when multiplied by X. Suppose a, b are real numbers that give the same product with X. Then by adding (-b)X to each side of the equality and factoring, we see (a-b)X = (b-b)X = 0, which implies a = b.

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Here are those handouts for your potential use:

• Proof-writing tips and notes on mathematical definitions, aimed at my linear algebra students but not subject-specific.
• An explanation, with examples and exercises, of mathematical induction.

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• Slides from the talk.
• Pdf about uses of linear algebra. From Oliver Knill, Harvard Math, near the bottom of this page, which has a large number of other handouts about linear algebra in general and applications.

Vector graphics

• Some Adobe Photoshop help. Unfortunately you have to go to the index and click V to find the vector graphics segment, but it is the source of the bicycle graphic in my slides and has a nice explanation.
• Interactive Mathematics’ vector art page, with explanation of more of the math behind the pictures and links to lots of other pages.
• Inkscape is a free drawing program that makes vector graphics natively. They have many useful tutorials and you can find even more elsewhere on the web.

Google PageRank

• Original PageRank paper
• PageRank example

Markov chains

• Fun With Markov Chains. Has examples where Alice in Wonderland was mixed together with either Hamlet or the Biblical books of Genesis and Revelation.
• Generating Text, Jon Bentley. Has examples where the King James Bible was remixed, and where his own text Programming Pearls was remixed.
  Note: the friend I corralled into compiling the C code for me said the first link’s program gave immense amounts of output without seeming to be near an end, whereas the second program gave much more reasonable files. I suspect based on the files I received that the amount of output in the second case is proportional to the amount of input. Reading the first page indicates you can specify the number of words of output in the command line; I don’t know if my friend did that.
• Random Word Generator
• Mark V. Shaney text generator

More details on the example in the slides above:
The Robert Frost poems included were Birches, Come In, Desert Places, Mending Wall, Nothing Gold Can Stay, On Looking Up by Chance at the Constellations, Reluctance, and Stars. The Dr. Seuss books included were One Fish, Two Fish, Red Fish, Blue Fish, and The Sneetches. The full recombined text may be found here, though I only put line breaks into the first third or so. The rest is unmined for Seufrostian gems. I tried putting a combination of the Dartmouth Marching Band song lyrics (only the songs related to Dartmouth, of course), and the standard description of Daniel Webster’s closing in the Dartmouth Case, but it did not turn up much of anything. If you don’t want to compile the C code in one of the first two links, Mark V. Shaney could be the answer. If it turns up any pearls let me know!

Principal component analysis

• Source of graphic in the slides.
• Chapter on PCA. Long, but a good example and overview at the start. This links directly to the pdf; I was unable to find a descriptive page for it.

Miscellaneous Fun

• The Onion

We’ll close with the only linear algebra joke I found worth sharing, a limerick by Donald E. Simanek.

Null vectors have zero projection.
So you ask, “What can be their direction?”
They point any which way.
“That’s magic!” you say?
Not really; it’s just misdirection.

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