Name: 改變基礎｜線性代數精髓，第13章。 (Change of basis | Essence of linear algebra, chapter 13)
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If I have a vector sitting here in 2D space

we have a standard way to describe it with coordinates.

In this case, the vector has coordinates [3, 2],

which means going from its tail to its tip

involves moving 3 units to the right and 2 units up.

Now, the more linear-algebra-oriented way to describe coordinates

is to think of each of these numbers as a scalar

a thing that stretches or squishes vectors.

You think of that first coordinate as scaling i-hat

the vector with length 1, pointing to the right

the vector with length 1, pointing straight up.

The tip to tail sum of those two scaled vectors

is what the coordinates are meant to describe.

You can think of these two special vectors

as encapsulating all of the implicit assumptions of our coordinate system.

The fact that the first number indicates rightward motion

that the second one indicates upward motion

All of that is tied up in the choice of i-hat and j-hat

as the vectors which are scalar coordinates are meant to actually scale.

Anyway to translate between vectors and sets of numbers

and the two special vectors, i-hat and j-hat, are called the basis vectors

is the idea of using a different set of basis vectors.

For example, let's say you have a friend, Jennifer

who uses a different set of basis vectors which I'll call b1 and b2

Her first basis vector b1 points up into the right a little bit

and her second vector b2 points left and up

Now, take another look at that vector that I showed earlier

The one that you and I would describe using the coordinates [3, 2]

Jennifer would actually describe this vector with the coordinates [5/3, 1/3]

What this means is that the particular way to get to that vector

In a little bit, I'll show you how you could have figured out those two numbers 5/3 and

In general, whenever Jennifer uses coordinates to describe a vector

she thinks of her first coordinate as scaling b1

What she gets will typically be completely different

from the vector that you and I would think of as having those coordinates.

To be a little more precise about the setup here

is something that we would describe with the coordinates [2, 1]

is something that we would describe as [-1, 1].

But it's important to realize from her perspective in her system

those vectors have coordinates [1, 0] and [0, 1]

They are what define the meaning of the coordinates [1, 0] and [0, 1] in her world.

So, in effect, we're speaking different languages

We're all looking at the same vectors in space

but Jennifer uses different words and numbers to describe them.

Let me say a quick word about how I'm representing things here

and so it depends on our choice of basis.

which would be an equally made-up construct

to help follow the meaning of her coordinates.

Her origin, though, would actually line up with ours

since everybody agrees on what the coordinates [0, 0] should mean.

will be different, depending on her choice of basis vectors.

How we translate between coordinate systems?

If, for example, Jennifer describes a vector with coordinates [-1, 2]

what would that be in our coordinate system?

How do you translate from her language to ours?

as they're represented in our coordinate system

you get a vector with coordinates [-4, 1]

So, that's how we would describe the vector that she thinks of as [-1, 2]

This process here of scaling each of her basis vectors

by the corresponding coordinates of some vector

with a matrix whose columns represent Jennifer's basis vectors in our language

In fact, once you understand matrix-vector multiplication

as applying a certain linear transformation

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改變基礎｜線性代數精髓，第13章。 (Change of basis | Essence of linear algebra, chapter 13)

perspective

figure

empathy

scale