Despite two linear algebra classes, my knowledge consisted of “Matrices, determinants, eigen something something”.

Why? Well, let’s try this course format:

- Name the course
*Linear Algebra*but focus on things called matrices and vectors - Teach concepts like Row/Column order with mnemonics instead of explaining the reasoning
- Favor abstract examples (2d vectors! 3d vectors!) and avoid real-world topics until the final week

The survivors are physicists, graphics programmers and other masochists. We missed the key insight:

**Linear algebra gives you mini-spreadsheets for your math equations.**

We can take a table of data (a matrix) and create updated tables from the original. It’s the power of a spreadsheet written as an equation.

Here’s the linear algebra introduction I wish I had, with a real-world stock market example.

## What’s in a name?

“Algebra” means, roughly, “relationships”. Grade-school algebra explores the relationship between unknown numbers. Without knowing x and y, we can still work out that (x + y)^{2} = x^{2} + 2xy + y^{2}.

“Linear Algebra” means, roughly, “line-like relationships”. Let’s clarify a bit.

Straight lines are predictable. Imagine a rooftop: move forward 3 horizontal feet (relative to the ground) and you might rise 1 foot in elevation (The slope! Rise/run = 1/3). Move forward 6 feet, and you’d expect a rise of 2 feet. Contrast this with climbing a dome: each horizontal foot forward raises you a different amount.

Lines are nice and predictable:

- If 3 feet forward has a 1-foot rise, then going 10x as far should give a 10x rise (30 feet forward is a 10-foot rise)
- If 3 feet forward has a 1-foot rise, and 6 feet has a 2-foot rise, then (3 + 6) feet should have a (1 + 2) foot rise

In math terms, an operation F is linear if scaling inputs scales the output, and adding inputs adds the outputs:

In our example, F(x) calculates the rise when moving forward x feet, and the properties hold:

## Linear Operations

An operation is a calculation based on some inputs. Which operations are linear and predictable? Multiplication, it seems.

Exponents (F(x) = x^{2}) aren’t predictable: 10^{2} is 100, but 20^{2} is 400. We doubled the input but quadrupled the output.

Surprisingly, regular addition isn’t linear either. Consider the “add three” function:

We doubled the input and did not double the output. (Yes, F(x) = x + 3 happens to be the equation for an *offset* line, but it’s still not “linear” because F(10) isn’t 10 * F(1). Fun.)

Our only hope is to multiply by a constant: F(x) = ax (in our roof example, a=1/3). However, we can still combine linear operations to make a new linear operation:

G is made of 3 linear subpieces: if we double the inputs, we’ll double the output.

We have “mini arithmetic”: multiply inputs by a constant, and add the results. It’s actually useful because we can split inputs apart, analyze them individually, and combine the results:

If the inputs interacted differently (like exponents, say), we couldn’t separate them — we’d have to analyze everything at once.

## Organizing Inputs and Operations

Most courses hit you in the face with the details of a matrix. “Ok kids, let’s learn to speak. Select a subject, verb and object. Next, conjugate the verb. Then, add the prepositions…”

**No!** Grammar is not the focus. What’s the key idea?

- We have a bunch of inputs to track
- We have predictable, linear operations to perform (our “mini-arithmetic”)
- We generate a result, perhaps transforming it again

Ok. First, how should we track a bunch of inputs? How about a list:

```
x
y
z
```

Not bad. We could write it (x, y, z) too — hang onto that thought.

Next, how should we track our operations? Remember, we only have “mini arithmetic”: multiplications, with a final addition. If our operation F behaves like this:

We could abbreviate the entire function as (3, 4, 5). We know to multiply the first input by the first value, the second input by the second value, etc., and add the result.

Only need the first input?

Let’s spice it up: how should we handle multiple sets of inputs? Let’s say we want to run operation F on both (a, b, c) and (x, y, z). We could try this:

But it won’t work: F expects 3 inputs, not 6. We should separate the inputs into groups:

```
1st Input 2nd Input
--------------------
a x
b y
c z
```

Much neater.

And how could we run the same input through several operations? Have a row for each operation:

```
F: 3 4 5
G: 3 0 0
```

Neat. We’re getting organized: inputs in vertical columns, operations in horizontal rows.

## Visualizing The Matrix

Words aren’t enough. Here’s how I visualize inputs, operations, and outputs:

Imagine “pouring” each input through each operation:

As an input passes an operation, it creates an output item. In our example, the input (a, b, c) goes against operation F and outputs 3a + 4b + 5c. It goes against operation G and outputs 3a + 0 + 0.

Time for the red pill. A matrix is a shorthand for our diagrams:

A matrix is a single variable representing a spreadsheet of inputs or operations.

**Trickiness #1: The reading order**

Instead of an input => matrix => output flow, we use function notation, like y = f(x) or f(x) = y. We usually write a matrix with a capital letter (F), and a single input column with lowercase (x). Because we have several inputs (A) and outputs (B), they’re considered matrices too:

**Trickiness #2: The numbering**

Matrix size is measured as RxC: row count, then column count, and abbreviated “m x n” (I hear ya, “r x c” would be easier to remember). Items in the matrix are referenced the same way: a_{ij} is the ith row and jth column (I hear ya, “i” and “j” are easily confused on a chalkboard). Mnemonics are ok *with context*, and here’s what I use:

- RC, like Roman Centurion or RC Cola
- Use an “L” shape. Count down the L, then across

Why does RC ordering make sense? Our operations matrix is 2×3 and our input matrix is 3×2. Writing them together:

```
[Operation Matrix] [Input Matrix]
[operation count x operation size] [input size x input count]
[m x n] [p x q] = [m x q]
[2 x 3] [3 x 2] = [2 x 2]
```

Notice the matrices touch at the “size of operation” and “size of input” (n = p). They should match! If our inputs have 3 components, our operations should expect 3 items. In fact, we can *only* multiply matrices when n = p.

The output matrix has m operation rows for each input, and q inputs, giving a “m x q” matrix.

## Fancier Operations

Let’s get comfortable with operations. Assuming 3 inputs, we can whip up a few 1-operation matrices:

- Adder: [1 1 1]
- Averager: [1/3 1/3 1/3]

The “Adder” is just a + b + c. The “Averager” is similar: (a + b + c)/3 = a/3 + b/3 + c/3.

Try these 1-liners:

- First-input only: [1 0 0]
- Second-input only: [0 1 0]
- Third-input only: [0 0 1]

And if we merge them into a single matrix:

```
[1 0 0]
[0 1 0]
[0 0 1]
```

Whoa — it’s the “identity matrix”, which copies 3 inputs to 3 outputs, unchanged. How about this guy?

```
[1 0 0]
[0 0 1]
[0 1 0]
```

He reorders the inputs: (x, y, z) becomes (x, z, y).

And this one?

```
[2 0 0]
[0 2 0]
[0 0 2]
```

He’s an input doubler. We could rewrite him to `2*I`

(the identity matrix) if we were so inclined.

And yes, when we decide to treat inputs as vector coordinates, the operations matrix will transform our vectors. Here’s a few examples:

- Scale: make all inputs bigger/smaller
- Skew: make certain inputs bigger/smaller
- Flip: make inputs negative
- Rotate: make new coordinates based on old ones (East becomes North, North becomes West, etc.)

These are geometric interpretations of multiplication, and how to warp a vector space. Just remember that vectors are *examples* of data to modify.

## A Non-Vector Example: Stock Market Portfolios

Let’s practice linear algebra in the real world:

- Input data: stock portfolios with dollars in Apple, Google and Microsoft stock
- Operations: the changes in company values after a news event
- Output: updated portfolios

And a bonus output: let’s make a new portfolio listing the net profit/loss from the event.

Normally, we’d track this in a spreadsheet. Let’s learn to think with linear algebra:

The input vector could be ($Apple, $Google, $Microsoft), showing the dollars in each stock. (Oh! These dollar values could come from

*another*matrix that multiplied the number of shares by their price. Fancy that!)The 4 output operations should be: Update Apple value, Update Google value, Update Microsoft value, Compute Profit

Visualize the problem. Imagine running through each operation:

The key is understanding *why* we’re setting up the matrix like this, not blindly crunching numbers.

Got it? Let’s introduce the scenario.

Suppose a secret iDevice is launched: Apple jumps 20%, Google drops 5%, and Microsoft stays the same. We want to adjust each stock value, using something similar to the identity matrix:

```
New Apple [1.2 0 0]
New Google [0 0.95 0]
New Microsoft [0 0 1]
```

The new Apple value is the original, increased by 20% (Google = 5% decrease, Microsoft = no change).

Oh wait! We need the overall profit:

Total change = (.20 * Apple) + (-.05 * Google) + (0 * Microsoft)

Our final operations matrix:

```
New Apple [1.2 0 0]
New Google [0 0.95 0]
New Microsoft [0 0 1]
Total Profit [.20 -.05 0]
```

Making sense? Three inputs enter, four outputs leave. The first three operations are a “modified copy” and the last brings the changes together.

Now let’s feed in the portfolios for Alice ($1000, $1000, $1000) and Bob ($500, $2000, $500). We can crunch the numbers by hand, or use a Wolfram Alpha (calculation):

(Note: Inputs should be in columns, but it’s easier to type rows. The Transpose operation, indicated by t (tau), converts rows to columns.)

The final numbers: Alice has $1200 in AAPL, $950 in GOOG, $1000 in MSFT, with a net profit of $150. Bob has $600 in AAPL, $1900 in GOOG, and $500 in MSFT, with a net profit of $0.

What’s happening? **We’re doing math with our own spreadsheet.** Linear algebra emerged in the 1800s yet spreadsheets were invented in the 1980s. I blame the gap on poor linear algebra education.

## Historical Notes: Solving Simultaneous equations

An *early* use of tables of numbers (not yet a “matrix”) was bookkeeping for linear systems:

becomes

We can avoid hand cramps by adding/subtracting rows in the matrix and output, vs. rewriting the full equations. As the matrix evolves into the identity matrix, the values of x, y and z are revealed on the output side.

This process, called Gauss-Jordan elimination, saves time. However, linear algebra is mainly about matrix transformations, not solving large sets of equations (it’d be like using Excel for your shopping list).

## Terminology, Determinants, and Eigenstuff

Words have technical categories to describe their use (nouns, verbs, adjectives). Matrices can be similarly subdivided.

Descriptions like “upper-triangular”, “symmetric”, “diagonal” are the shape of the matrix, and influence their transformations.

The **determinant** is the “size” of the output transformation. If the input was a unit vector (representing area or volume of 1), the determinant is the size of the transformed area or volume. A determinant of 0 means matrix is “destructive” and cannot be reversed (similar to multiplying by zero: information was lost).

The **eigenvector** and **eigenvalue** represent the “axes” of the transformation.

Consider spinning a globe: every location faces a new direction, except the poles.

An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix.

(My intuition here is weak, and I’d like to explore more. Here’s a nice diagram and video.)

## Matrices As Inputs

A funky thought: we can treat the operations matrix as inputs!

Think of a recipe as a list of commands (*Add 2 cups of sugar, 3 cups of flour…*).

What if we want the metric version? Take the instructions, treat them like text, and convert the units. The recipe is “input” to modify. When we’re done, we can follow the instructions again.

An operations matrix is similar: commands to modify. Applying one operations matrix to another gives a new operations matrix that applies *both* transformations, in order.

If N is “adjust for portfolio for news” and T is “adjust portfolio for taxes” then applying both:

TN = X

means “Create matrix X, which first adjusts for news, and then adjusts for taxes”. Whoa! We didn’t need an input portfolio, we applied one matrix directly to the other.

The beauty of linear algebra is representing an entire spreadsheet calculation with a single letter. Want to apply the same transformation a few times? Use N^{2} or N^{3}.

## Can We Use Regular Addition, Please?

Yes, because you asked nicely. Our “mini arithmetic” seems limiting: multiplications, but no addition? Time to expand our brains.

Imagine adding a dummy entry of 1 to our input: (x, y, z) becomes (x, y, z, 1).

Now our operations matrix has an extra, known value to play with! If we want `x + 1`

we can write:

```
[1 0 0 1]
```

And `x + y - 3`

would be:

```
[1 1 0 -3]
```

Huzzah!

Want the geeky explanation? We’re pretending our input exists in a 1-higher dimension, and put a “1” in that dimension. We *skew* that higher dimension, which looks like a *slide* in the current one. For example: take input (x, y, z, 1) and run it through:

```
[1 0 0 1]
[0 1 0 1]
[0 0 1 1]
[0 0 0 1]
```

The result is (x + 1, y + 1, z + 1, 1). Ignoring the 4th dimension, every input got a +1. We keep the dummy entry, and can do more slides later.

Mini-arithmetic isn’t so limited after all.

## Onward

I’ve overlooked some linear algebra subtleties, and I’m not too concerned. Why?

These metaphors are helping me *think* with matrices, more than the classes I “aced”. I can finally respond to “Why is linear algebra useful?” with “Why are spreadsheets useful?”

They’re not, unless you want a tool used to attack nearly every real-world problem. Ask a businessman if they’d rather donate a kidney or be banned from Excel forever. That’s the impact of linear algebra we’ve overlooked: efficient notation to bring spreadsheets into our math equations.

Happy math.

## Other Posts In This Series

- A Visual, Intuitive Guide to Imaginary Numbers
- Intuitive Arithmetic With Complex Numbers
- Understanding Why Complex Multiplication Works
- Intuitive Guide to Angles, Degrees and Radians
- Intuitive Understanding Of Euler's Formula
- An Interactive Guide To The Fourier Transform
- Intuitive Understanding of Sine Waves
- An Intuitive Guide to Linear Algebra
- A Programmer's Intuition for Matrix Multiplication

## Leave a Reply

82 Comments on "An Intuitive Guide to Linear Algebra"

A spreadsheet is actually a much more general structure than a vector space, but leaving that aside, there are two insights being mashed into one here:

1. Many problems, though not obviously geometric, can be shoved into some geometric form and more easily tackled thus.

2. There is a scatter of algebraic structures describing features of various geometries that have proved useful over the years. The scatter is roughly a cone from the origin of unstructured sets out to things like noncommutative geometry.

There are other things near to vector spaces in that scatter, like affine spaces (vector spaces without origin) and modules (vector spaces over rings instead of fields), and all kinds of fascinating things that show up (look up tropical semirings and you’ll go down an enormous rabbit hole, all without ever leaving familiar algebraic operations). Reach a little farther out and you find yourself with inner product spaces, then add some calculus and you get Banach spaces, and a little farther in Hilbert space (weird fact: there’s only one Hilbert space; all its various appearances are all isomorphic). Keep going and you start finding yourself in Riemannian geometry, and then even farther out in noncommutative geometry and the like.

Vector spaces are a sweet spot, for three reasons:

1. They’re sufficiently unstructured where most of the components of more complicated geometries like tangent spaces and the like will all be vector spaces.

2. They’re just enough structure to always be able to express any abstract vector space in familiar vectors of numbers and operations on them as matrices. Thus you can always grab a basis and start computing, no matter how exotic your vector space may seem.

3. They’re turtles all the way down. The space of linear operations on a vector space is a vector space. The space of coordinate transformations from one basis on a vector space to another basis is a vector space. When you start adding inner products and the like, you can pretty well always find a way of looking at them where they’re just another vector space.

They have oddly nice properties as well. For example, no matter how weird the vector space, it has a well defined dimension (though it may be infinite).

Linear algebra “done right” is really a question about the structure that emerges from a very broad class of geometric problems. The really interesting part is how you suture vector spaces together in various ways to get other classes of geometries entirely. For instance, a curved space isn’t a vector space, but define a tangent space at every point of that space. The tangent space at a point is a vector space with the same dimension as the space. You can think of it as the velocity of an object at that point. The geometry comes in when you use the notion of tangents as velocities to map back to actual paths in the space.

So matrices and vectors of numbers are nice, but they’re barely the tip of the ice berg of linear algebra.

I think you’re glazing over the main point of matrices:

Every linear map can be represented by a matrix.

This should not be obvious to the beginning student. We don’t work with matrices just because they give us a useful way to organize information, because that simply wouldn’t be useful if we couldn’t use them to represent any linear map. We work with matrices because they completely characterize the functions we care about.

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

Hello

Im a begginer learning algebra. Excuse my is my question is absurd.

When you present the operation displaystyle{F(x, y, z) = 3x + 4y + 5z}

Why do we sum their results?

I mean.. we did summed in the cases you presented earlier because the opration was the same for all inputs, like in

displaystyle{G(x, y, z) = F(x + y + z) = F(x) + F(y) + F(z)}

But {F(x, y, z) = 3x + 4y + 5z} is composed of three different operations, could be like {F(x, y, z) = A(x) +B(y)+ C(z)}

I understand that we combine linear operations (A(x), B(y), C(z)) to create F(x), but how this one respect the rule “adding inputs adds the outputs”?

Thanks a lot!

I like to think of Eigenvectors and eigenvalues from the perspective of a pitcher standing on the mound on a windy day. In the space between the pitcher and home plate there are many lines of force going in different directions created by the swirling winds. Some of these happen to line up perfectly with the direction the pitcher is throwing the ball. These are Eigenvectors. When the ball is thrown along this perfectly lined up vector it will gain speed in addition to the initial speed it was thrown with. This additional speed is the eigenvalue. The lines of force are contours in the space that influence the way the ball moves through that space.

I have to disagree on the “spreadsheet” approach to linear algebra. Matrix/vector multiplication never made any sense to me, until I realized it’s just projecting the vector onto the original identity basis, and then reconstituting it using the new basis instead. You can discover and draw this process entirely visually.

The relationship between a matrix and the vectors made up of its rows or columns is ridiculously obvious once you see it in action. Yet in years of linear algebra and engineering, nobody ever bothered to show this to me.

Like most things the best way of learning something is to approach it from different viewpoints. This article does that although not convinced about keeping examples abstract.

I only really understood the advantages of a Matrix when I had to write a program to rotate points in 3d space on a different course. Did the rotation equations then the whole concept of matrix maths ‘clicked’ when I realised its nothing special it’s just a neat way of doing the maths!

In fact the whole mystery of maths clicked – maths is nothing more than a human language and tool for describing how things interact. Maths doesn’t have rules it simply implements observations of physical reality in a convenient way. For example complex numbers in Electrical engineering integration etc.

Of course the discovered rules can then hint at other physical rules that haven’t yet been discovered, which is the real power of maths in general!

Great choice of topic! I jumped on this one hoping to refresh my memory on linear algebra and reconsider its usefulness, but in this article it was a bit of a bumpy ride. I would say your usual style allows for a much smoother transition from building blocks to a-ha moments.

In “Organizing Inputs and Operations”, if you look carefully, this will read smoothly to someone who already understands what you’re talking about, but a novice would be lost. You introduce two different operations at the same time as you’re explaining what the rows mean in the matrix notation, leading to both points being hard to catch. Going forward, you frequently forget you’ve not introduced a notion before you start using it (“transformation”), using the “axes” in the globe analogy without really explaining what you’re doing there etc.

I hope you take it well – this article definitely is better explained than what I got in college, but I came to expect even better from you :)

I wrote some of what I know about eigenvalues and determinants down here: http://ajkjk.com/blog/?p=18

Maybe it will be helpful for intuition.

could you please repost the link ? thx alot

Thanks for sharing your insights on matrices…reading this brought a tear to my eye…I wish my school teachers were like you.

[…] EDIT: Note that this series is not intended to teach linear algebra, but to record insights into visualisation aids for linear algebra for those who are already in the process of learning from some other source. If you’d like to actually learn linear algebra, there are other great resources like this one. […]

Thank you so much for this – having read and used your imaginary numbers post in my recent introduction for year 12, I thought ‘I wonder if he’s done anything on matrices?’ Watched Derek Holt’s lectures on Linear Algebra over the summer (they’re very good), but for a really intuitive introduction I can’t ask for better than this page. Granted, it won’t be a full description, but what I really need is an intuitive hook to get started with, and this was definitely it! My year 12 Further Maths group thank you!

Awesome, really glad to hear it helped :). Linear algebra befuddled me for a while because I always associated with “advanced” operations, like rotating a robotic arm in 3d or solving a giant system of equations. No — we can just take an everyday example, like having a stock portfolio and updating it based on some event. Seeing it as a ‘mini spreadsheet’ helped me wrap my mind around the use cases, which can of course expand into to the fancy vector operation stuff.

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.

“Guassian elimination” typo

[…] is in reply to this linear algebra guide, which came up on Hackernews today. The author wrote that he didn't have a good intuition about […]

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“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… :)

Actually computer programs existed before computers, and I think as well he underestimates solving multiple equations with matrices (that is done all the time in science, e.g. for modelling data).

but his point is still correct, the linear algebra is wasted due to the way of explanation and still is.

The way they teach school or university up till today and all the online courses was exactly about knowing, sizes, vectors, positive definite and determinant zero. Those things that you always end why? and now I am working with tensors not knowing why they said some times skewing. I never saw what was skewed.

When I printed this, the first letter of every line was cut off. Can you fix the print css?

thank you very much!)

… 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.

[…] An Intuitive Guide to Linear Algebra — Here’s the linear algebra introduction I wish I had. I wish I’d had it, too. (via Hacker News) […]

@Tom Elovi Spruce,

I believe, the topic of the article at hand is math, as opposed to literature, but if it were, you’d have a valid point.

[…] An Intuitive Guide to Linear Algebra You need to understand matrices for data mining, machine learning, and other advanced computer science – so if you happen to forget what you learned in a college (“the survivors are physicists, graphics programmers and other masochists”) this is a good basic primer. […]