All of Linear Algebra in Five Minutes

Thursday October 15, 2026

I volunteered to give a talk on Linear Algebra at the Data Science DC lightning talks event today. I may be able to get it recorded; for now the full deck is available. It is challenging to present all of Linear Algebra in five minutes. I hope it's fun!

Script

This won't make a lot of sense without the slides; check out the deck, where the script is all in the speaker notes. (I wrote and edited it in here originally though, so I might as well keep it, I figure.)

It arguably still doesn't make enough sense with the slides, because of (among other possible reasons) the constraints of fitting into a five minute presentation. I am working on more complete explanations; see Linear Algebra project.

If math is about patterns, what patterns is Linear Algebra about?

Well, patterns like this: linear ones.

Let's fill in the gap.

We can add existing entries.

We can also multiply existing entries.

And there's a pattern: times two.

We'll call that two a “singular value” and also an “eigenvalue” here.

In one dimension, this is really all we can do.

Proof.

So what is this?

Multiplication emits multiples, but it also detects, like a measuring stick with different units.

And you can go either way, to the right or the left, and each is the inverse of the other. Two is “invertible.”

The unit, one, is sort of special, just sort of cosigning whatever you already had.

But zero sends everything to a single value. Zero is singular. There's no inverse, “you can't divide by zero.”

Negative numbers work fine though.

Or we can separate the size from the sign, and get the singular value decomposition of negative two.

Then it's the usual positive multiplication of lengths, and for the signs, either they cosign or they don't.

Also now we have two unit measuring sticks.

What if we have two inputs and one output? All we can do is this.

Proof.

We'll also call this kind of thing a dot product.

Let's write it differently.

Let's put the a and b like this.

Which we'll call a row.

And to be consistent with function notation, we'll say the two inputs come on the right.

So the inputs have to come like this.

So we better write them like this.

And we'll call that a column.

The choice of writing it this way is arbitrary, so it deserves a mnemonic.

“Rows reach right, columns lean left.”

Now this is all lined up like a pipeline. If we flip that pipeline...

It has to come out like this, with rows and columns transposed.

Okay, call this guy a vector. How long is it? Maybe you know the Pythagorean theorem?

Proof.

The square root isn't linear, but we can use it to rewrite things. We get essentially the singular value decomposition of a vector: a length times a unit direction. This is basically polar coordinates.

All the 2D unit vectors make this circle. And all of them are great measuring sticks.

They measure how long a vector is, in their direction. For two unit vectors, that's no-kidding cosine, and it's invariant to rotation.

Proof.

When things are perpendicular, the dot product and the cosine are zero.

So vectors are detectors and emitters.

Here's a unit vector.

It measures a vector coming on the right.

Or it emits a multiple of itself.

A column is just the transpose.

It measures a vector coming on the left.

Or it emits a multiple of itself. (This is the one 3Blue1Brown focuses on.)

This is the mnemonic.

Let's do multiple inputs and multiple outputs.

Here's an orthogonal matrix, perpendicular unit vectors. On the left, it detects along the row directions and emits along the column directions.

The transpose does the opposite, so it's the inverse.

And we already did inverses for diagonal matrices.

There might be some zeros.

At last we have the full Singular Value Decomposition, SVD, the Fundamental Theorem of Linear Algebra! Every matrix can be written like this!

Notice rows on the right, columns on the left.

That's also PCA, by the way. [Principal Components Analysis]

And you already know how to invert an SVD, or get close.

So you can solve equations like this. But use a computer.

And it won't work if the matrix is singular.

But you can still get some sort of solution like this. And that's linear regression.

Data though. Kids rated foods.

It's nice to subtract out the means.

Maybe you can see there's only one singular value, and we could do dimensionality reduction.

Measure columns by themselves, you get sums of squares, a multiple of the covariance matrix.

We can make the rows unit length.

And measure rows by themselves. These are cosines, aka Pearson's correlation coefficients.

These Gram matrices are always square. People really like square matrices.

With a square matrix, you can find eigenvectors that just get stretched by an eigenvalue, without changing direction. Maybe you can diagonalize a matrix.

This lets you do fun stuff like getting the closed form for the nth Fibonacci number.

And that's about it! Thanks!