Basis, vector space, and linear independence

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From the course by Imperial College London

Mathematics for Machine Learning: Linear Algebra

904 оценки

Imperial College London

Mathematics for Machine Learning: Linear Algebra

904 оценки

Course 1 of 3 in the Specialization Mathematics for Machine Learning

In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning.

From the lesson

Vectors are objects that move around space

In this module, we look at operations we can do with vectors - finding the modulus (size), angle between vectors (dot or inner product) and projections of one vector onto another. We can then examine how the entries describing a vector will depend on what vectors we use to define the axes - the basis. That will then let us determine whether a proposed set of basis vectors are what's called 'linearly independent.' This will complete our examination of vectors, allowing us to move on to matrices in module 3 and then start to solve linear algebra problems.

Changing basis11:24

Basis, vector space, and linear independence4:18

Applications of changing basis3:28

Meet the Instructors

David Dye
Professor of Metallurgy
Department of Materials
Samuel J. Cooper
Lecturer
Dyson School of Design Engineering
A. Freddie Page
Strategic Teaching Fellow
Dyson School of Design Engineering

[MUSIC]

So we've seen how we don't just have to have basis vectors that are our normal 1,0 and

0,1, the so-called natural basis here.

We can have different basis vectors that redefine how we we move about space.

In this video, what we're going to do is we are going to define what we mean by

a basis, by a vector space, and by the term linear independence, which

is going to let us understand how many dimensions our vector space actually has.

So first let's define what we mean by a basis.

A basis is a set of n vectors that are not linear combinations of each other,

which means they're linearly independent and

that they span the space that they describe.

The space is then n-dimensional.

The first criteria I can write down by saying that if they're

linearly independent,

then I can't write any of them by taking some combination of the others.

For example, if I had a candidate, b3,

it would be linearly dependent on the basis vectors b1 and b2,

if I could take some combination of them a1b1s + a2b2s,

where a1 and a2 are numbers, if I could take a linear

combination of b1 and b2 like this, and find b3,

then, b3 would be linearly dependent on b1 and b2.

It wouldn't be an independent vector, so it's some vector.

I can sketch it out by doing something like this.

If I've got b1 here, and say b2 here, if b3, I can find by

some combination of b1 and

b2s, it must lie on the same plane as b1 and b2, all right, the plane of my board.

If b3's actually lying out of the plane here,

I couldn't make the out-of-plane component of it using my b1s and b2s.

So b3 would then be linearly independent of b1 and b2.

This = I couldn't ever make true, and then b3 would lie outside of the plane.

I'd then have three basis vectors.

Those three would then define a three dimensional vector space, and

I could get anywhere in 3D space with them.

And then if I had another basis vector, b4, that wasn't a linear combination of

b1, b2, and b3, then b4 would then let me have a four-dimensional space and

so on and so on to as many dimensions as I like.

Then a vector is in the vector space spanned by b1 and

b2 if it can be found by a linear combination of b1s and b2s.

Now, notice what my basis vectors b don't have to be.

They don't have to be unit vectors, by which I mean vectors of length 1.

And they don't have to be orthogonal, at 90 degrees to each other,

normal to each other.

But everything is going to be much easier if they are.

So if at all possible,

you want to construct what's called an orthonormal basic vector set

where they're all at 90 degrees to each other and all of unit length.

Now, let's think about what happens when we map from one basis to another.

The number line of the axes of the original grid

then projects down onto the new grid, like here.

And potentially has different values on that grid, but

the projection keeps the grid being evenly spaced.

Therefore, any mapping that we do from one set of basis vectors,

one coordinate system to another set of basis vectors, another coordinate system,

keeps the vector space being a regularly spaced grid, where

our original rules of vector addition and multiplication by a scaler still work.

It doesn't warp or fold space,

which is what the linear bit in linear algebra means geometrically.

Things might be stretched or rotated or inverted, but

everything remains evenly spaced and linear combinations still work.

Now, when the new basis vectors aren't orthogonal, then to do the change from one

base to another, we won't just be able to use the dot product anymore.

We'll have to use matrices instead, which we'll meet in the next module.

So that's the formal definition of what we mean by a basis and

by linear independence.

[MUSIC]

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