Bad Teachers Destroy The Beauty of MATHS!（原声文稿）

You might think matrices are just complicated mathematical objects with little practical use, but that's not true at all.

Matrices are extremely useful, and to understand why, let's start with a simple example.

Imagine I'm hosting a party with three guests, and each guest needs two eggs to prepare the batter. How many eggs should I buy?

It's straightforward. Multiply three guests by two eggs per person. and we get six eggs.

So, I need six eggs in total.

Of course, preparing crepes requires more than just eggs.

Let's say each person needs two eggs, one kilogram of flour, one liter of milk, and one bottle of beer for the batter.

We can organize these requirements in a column, with each row representing a different ingredient.

Since we have three guests, we multiply every requirement by three.

That gives us six eggs, three kilograms of flour, three liters of milk, and three bottles of beer.

This is our shopping list for the party, but let's make the example a little more realistic.

Not everyone eats the same way, so imagine that my party includes three regular guests, two rugby players, and four prima ballerinas.

Each category has different ingredient requirements.

A A regular guest needs two eggs, a rugby player needs four eggs, and a prima ballerina needs three eggs.

To calculate the total number of eggs, we multiply the number of people in each category by the number of eggs required per person.

For the regular guests, we need three multiplied by two, which gives six eggs.

For the rugby players, we need two multiplied by four, which gives eight eggs.

And for the prima ballerinas, we need 4 multiplied by 3, which gives 12 eggs.

Now, we add these amounts together, 6 plus 8 plus 12 equals 26 eggs.

So, we need to buy 26 eggs for the entire party.

Now that we understand the idea, let's calculate the requirements for all 4 ingredients and all 3 categories of guests.

We can organize this information into a rectangular table of numbers called a matrix.

Each column represents one category of people.

The first column represents regular guests.

The second column represents rugby players. And the third column represents prima ballerinas.

Each row represents a different ingredient, eggs, flour, milk, and beer.

Notice that the entry in the beer row for the prima ballerinas is zero.

This simply means that the prima ballerinas don't need beer for their batter, so we don't need to buy any beer for that category.

Every number in the matrix tells us how much of a particular ingredient one person in that category requires.

We then multiply each column by the number of people in its corresponding category.

The first column is multiplied by three, because we have three regular guests.

The second column is multiplied by two, because we have two rugby players.

And the third column is multiplied by four, because we have four prima ballerinas.

After multiplying each column, we add the numbers across each row to find the total quantity of each ingredient.

When you see this calculation written mathematically, Mathematically, it may look like a complicated operation involving a matrix and a vector, but it becomes much easier to understand when you picture how the information flows.

The matrix is the table containing the requirements, and the vector on the right contains the number of people in each category.

Each number in that vector tells us how many times to multiply the corresponding column of the matrix.

We then add the results across each row to obtain the final quantities.

There is one particularly important detail to remember. The order of the columns matters.

The first column represents regular guests. The second represents rugby players.

And the third represents prima ballerinas.

Each coordinate matches the corresponding column.

This is how we decide what information belongs in each column of a matrix.

Now, let's look at this calculation from a more general mathematical perspective.

We can think of the entire process as a function.

The input is the vector containing the numbers of people in each category.

The matrix acts as the function, taking that input and producing the output vector containing the quantities of ingredients we need.

In other words, we put three numbers into the function and get four numbers out.

out. Why three inputs and four outputs?

Because there are three categories of guests, so the input vector has three coordinates, and there are four ingredients, so the output vector has four coordinates.

This also explains the dimensions of our matrix.

It has three columns, corresponding to the three inputs and four rows, corresponding to the four outputs.

The function represented by this matrix has two very interesting properties.

The first is related to proportionality.

Suppose I double the number of guests in every category.

If the number of regular guests, rugby players, and prima ballerinas all doubles, then every ingredient requirement doubles as well.

If I need 26 eggs for the original group, I'll need 52 eggs for twice as many people.

The same principle applies to flour, milk, and beer. Doubling the input doubles the output.

More generally, if we multiply the input vector by any number, the output vector is multiplied by that same number.

The second property appears when we have two separate groups of guests.

Suppose one group is represented by the vector x1, and another group is represented by the vector x2.

If we want to host both groups together, we simply add their numbers in each category.

The shopping list for the combined group is then the sum of the shopping lists for the two groups separately. This property is called additivity.

These two properties may look abstract when written using mathematical symbols, but the shopping example makes them intuitive.

Both properties follow naturally from the way a matrix performs its calculation.

A function that satisfies these two properties is called a linear function.

More precisely, it must satisfy both scaling and additivity.

Linear functions and matrices appear in many different fields, including mathematics, geometry, physics, and economics.

In geometry, for example, matrices can represent transformations such as stretching, rotating, shearing, and projecting shapes.

These transformations can happen in two dimensions or three dimensions.

The diagrams you see here illustrate how a shape can be stretched, rotated, sheared, or projected using a matrix.

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