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Speaker 1

00:01

Hello, I'm Sheldon Axler, the author of Linear Algebra Done Right. This video discusses the section of the book titled, Definition of Vector Space. Reminder, throughout these videos, F denotes either the field of real numbers R or the field of complex numbers C. The motivation for our definition of a vector space comes from properties of addition and scalar multiplication in Fn.

Speaker 1

00:30

Let's list those simple properties. Addition in Fn is commutative, associative, and has an identity. Every element of Fn has an additive inverse. Scalar multiplication in Fn has the appropriate associative property.

Speaker 1

00:50

Scalar multiplication by 1 in Fn acts as expected, meaning that 1 times a vector equals that vector. And finally, addition and scalar multiplication are connected by distributive properties.

Speaker 2

01:07

The properties of addition and scalar multiplication in Fn listed in these bullet points are precisely what we will use to get our definition of a vector space. We need to define what we mean by an addition and by a scalar multiplication on a set V. By an addition on a set V, we mean a function that assigns an element u plus w in V to each pair of elements u comma w in V.

Speaker 2

01:34

By a scalar multiplication on a set V, we mean a function that assigns an element lambda u in V to each lambda in F, our scalar field, and to each element u in v. Obviously, addition and scalar multiplication on fn give an example. Let's look at another example. Suppose v is a set of real valued functions on the closed interval 0, 1.

Speaker 2

02:00

For functions f and g in V and a real number lambda, we define the addition f plus g and the scalar multiple lambda f as you would expect. Namely, f plus g is the function whose value at x is f

Speaker 1

02:15

of x plus g of

Speaker 2

02:16

x, and lambda f is the function whose value at x is lambda times f of x. The key point here is that f plus g and lambda times f are both real valued functions defined on the interval 0, 1. Thus they are elements of V.

Speaker 2

02:35

Now we are ready to give the formal definition of a vector space. A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold. These properties are modeled on the corresponding properties of Fn. The first property is commutativity of addition, which means that the order in which we add elements does not matter.

Speaker 2

03:01

Next, we have associativity. First for addition, which means the grouping does not matter, and second for scalar multiplication, where we have 2 scalars and 1 element of V, and we can do the operations in either order, as indicated, and get the same result. Next, additive identity. There exists an element of V called 0, such that when we add it to any other element, we get the element we started with.

Speaker 2

03:30

Every element should have an additive inverse, meaning that we can add something to it and get 0, the additive identity. The scalar 1 works as we expected, meaning that 1 times any element of v gives us that element of v. And finally, we have 2 distributive properties. Notice they're slightly different.

Speaker 2

03:50

1 of them involves 1 scalar and 2 elements of v, and the other 1 involves 2 scalars and 1 element of v. But in both cases, we can distribute multiplication over addition as expected. Let's look at some examples of vector spaces. Our first example is Fn with the usual operations of addition and scalar multiplication.

Speaker 2

04:13

It's no surprise that Fn is an example of a vector space because we based our definition of vector space on the properties of fn. So let's look at another example. F infinity is defined to be the set of all sequence of elements of f. Addition and scalar multiplication on F infinity are defined just as you would expect.

Speaker 2

04:37

To add 2 sequences, we add them coordinate-wise, and to multiply a sequence by a scalar, we multiply coordinate-wise, as shown here. With these definitions, f infinity becomes a vector space, as you should verify. Let's look at another example of a vector space. Suppose f is a non-empty set.

Speaker 2

05:00

Let f to the power s denote the set of functions from s to f. We define the sum of 2 functions from s to f in the obvious way, namely, the value of the function f plus g at x is the value of f of x plus the value of g of x. Similarly, we define scalar products in the obvious way. If lambda is a scalar, meaning a real or complex number, and f is a function from s to f, we define lambda f to be the function from s to f whose value at any point x in S is lambda times f of x.

Speaker 2

05:37

With these definitions of addition and scalar multiplication, you should verify that f to the s power is a vector space. Now we come to the first theorem, whose proof we will discuss in these videos. This theorem states that if v is a vector space, then 0 times u equals 0 for every vector u in v. Notice the use of the word vector here.

Speaker 2

06:02

We frequently use the words vector and point to describe elements of V, but these words have no formal meaning. So vector is just shorthand for an element of V. Let's look at the proof of this theorem. For u, a vector, and v, we have 0 times u equals 0 plus 0 times u.

Speaker 2

06:22

That's because we're dealing with the number 0, which is equal to 0 plus 0. Now we use the distributive property to write the right side of that equation as 0 times u plus 0 times u.

Speaker 1

06:36

Next, we add the additive inverse of 0 times u to both sides of

Speaker 2

06:40

the equation above, getting the new equation shown here.

Speaker 1

06:45

Notice that on the left side of this equation, we have 0 times u plus the additive inverse of 0 times u, which of course adds up to 0. On the right side of

Speaker 2

06:54

the equation, we have 0 times u plus 0 times u plus the additive inverse of 0 times u. Grouping the last 2 terms together, we just get 0. And then adding that to 0 times u, we get 0 times u, giving us finally the equation 0 equals 0 times u, completing the proof.

Speaker 2

07:17

Let's conclude this video with a brief discussion of the advantages of the abstract approach to vector spaces. Abstraction has 2 main advantages in mathematics. The first advantage is that the abstract approach can lead 1 to apply what's been done to multiple new situations that may not have been anticipated previously. The second big advantage of abstraction is by stripping away inessential properties, 1 can see what's really going on, and this often leads to greater understanding.

Speaker 2

07:53

Finally, 1 comment and warning. If V is a vector space, it would be incorrect to prove that 0 times u equals 0 for every u and v by starting out as follows. Let u equal x1 up to xn, thus 0 times u is, and so on. That's incorrect because we're dealing with an abstract vector space and u might not be

Speaker 1

08:18

of the form a list of numbers.

Speaker 2

08:19

It might be some other object, it might be a function, something more abstract. Thus, the proof that we gave in the previous set of slides is the correct way to do that. We need not look at specific coordinates.

Speaker 2

08:33

To be specific and to repeat, an element of a vector space V is not necessarily of the form x1 comma dot dot dot xn, although of course elements

Speaker 1

08:42

of Fn are of that form.

Speaker 2

08:46

This concludes the video on definition of vector space.