Inner Prod Norm Pt. 1

Created Monday 08 June 2020

This is the fourth video our professor has uploaded.

We start the video by continuing our discussion of linearly independent vectors.

As a recap, we have two linearly independant vectors if the only way to add them to get a 0 vector is when their scalar coefficient are 0 as follows:

Definition: Let v₁, v₂, ..., v_k be vectors in V. We call the linear span of these vectors:
lin. span. {v₁, ..., v_k} =

Then the linear span {v_1, v_{2, ...,} v_k} is a vector space.

Let ,

linear span of {} =

How about an example that is not ? Observe:
Let v₁ = (1, 3) and v₂ = (-1/3, -1)

Notice how v₁ and v₂ are multiples of eachother!

Definition: If are vectors of space V such that:

1. are linearly independent
2. linear span = V

=> we call a basis of V and we call the number k the dimension of V and we write dimension of V = k. In otherwords, the number of vectors we have in our linear span is the number of dimensions we have for our basis.

dim V = k

We know that the vectors (1, 3) and (0, 1) are linearly independent. But how do we check this?
Let
We need to prove that both lambda and mu are 0.
So, we have this:

We such, we proved that these two vectors are linearly independent.

Remember, whenever we want to prove something we go immediately to the definition. Then we see if it applies to the problem we're working on. You will need to know your theory very well to make it through exercises.

Now then, why can any be written as a linear combination of (1, 3), (0, 1)?

Find

Now we can solve for the lambdas!

With this, we can write (1, 2) as a linear combination of the vectors:

Considering this, we have the following:

Since we can write any other vector from this two vectors, we have a basis.

The diagram above shows basis vectors in action.

This applies to any dimension! 1D, 2D, 3D, 4D, 5D, ..., nD.
Linearly independent vectors are basis vectors.

More on Basis Vectors

For , (1) is a basis vector and so is (-10) for any other vector can be written as a multiple of it. For example,
However, (1) is known as the natural basis.

For , we use two vectors: and

Likewise, we also have basis vectors for like so:

Finally, we have


...

This all make up the natural basis in

And Even More on Basis Vectors...

The following is a linear combination of the basis vectors:

If is a basis for V then any vector is written as a linear combination (sum):
I couldn't quite tell if I got that formula written correctly. The camera is very blurry.

Example of Vector Space Where It Cannot be Written Using Coordinates

{all continuous functions }
This reads that we have a set C and continuous function f from a to b that maps onto R.

We want to prove that C is a vector space. How do we do this?
For addition +, we have functions f and g. such that

For example:
We have the function.
which read the 0 vector plus any function f is equal to f.

What about the additive inverse?

We also know it is commutative and associative in regards to addition

What about scalar multiplication?

So, that works just fine as well!

So, we have a vector space. But! And then he says we can't have a set of something. I couldn't make out what he was referring to.
He goes on to say that we have an inifinite number of basis.

Inner Product or Dot Product

Yay my favorite!

Definition: The inner product is denoted by < , > : V * V
This reads "< , > is a function that..."
Also, is the inner product! The dots are placeholders for numbers we input.
< , > is not an operation though.

So, we have the following:

1. it is = 0 only if
2. which means it's symmetric.
3. Most importantly, the inner product is bilinear.

What do we mean by bilinear? Observe:

By the way, due to symmetry we can combine <v, w> and <w, v> as seen in the equation above.

If you have
The av + bw is linear and so is the z. Since both are linear, we having something that is bilinear.

Continuing on...

In the inner product:
Suppose and

For the following, we take the product of the corresponding coordinates.

Here is an example of what we just did but with actual numbers:

Furthermore:

The inner product is 0 only if a and b are both equal to 0.

Likewise,

We can make this more general as well. For example:
For let
When finding the inner product, we get the following:

Norm

Next, what is a Norm in a vector space?
Let Y be a vector space.
The norm is such that:

1. The norm of any vector v is positive. Also can be written as: .
   1. It is only equal 0 if
2. 
3. this is known as the triangle inequality.

In , the absolute value || satisfies all 3 of the above so in the vector space the absolute value |*| measures length or distance between points! (The * is just a standin for some value).

We can check that in

Knowing this, we get something similar to that of the Pythagorean Theorem:

So then, let's say we want the length of (a, b). Then we do this:

Let the following be a vector space:

Prove that

has also a norm

We have to check if this is a norm in .

As we can see, the norm and inner product can be applied to any vector space!

The video ends here.