Transcript:

In this video, we’ll define the norm and state some properties of it. We are still looking at column vectors in R^n and the norm of such a vector is written like this kind of two nested absolute value symbols, And it’s the square root of the dot product of the vector with itself; and because this is always greater than or equal to zero, The norm is always defined. Your intuition here should be that. The norm of a vector represents its length. Like if you have the vector [4; 2] and you want to know the length of the vector. Once you create a right triangle. One side is four. The other side is two, and the Pythagorean theorem says that the length squared is the first side squared, plus the second side squared. So the length is the square root of four squared, plus two squared and this is indeed the dot product of the vector with itself. Once we understand the norm as the lengths, some properties that the norm has should make sense. The first property is the famous triangle inequality that the norm of (u plus V) is less than or equal to (the norm of U) plus (the norm of V). And if you think of u and V as sitting tip to tail, then they are two sides of a triangle, and this inequality is telling you that one side of a triangle is less than or equal to the sum of the other sides, which is a well known geometric fact. Bearing in mind that the norm is the length and multiplying by a scalar just multiplies the length by the absolute value of the scalar that second property should make sense And certainly the length of any vector is greater than or equal to zero, and the only way for the length of a vector to equal zero is if the vector is the zero vector. Let’s end with a further example, not example. Definition: A unit vector is a vector whose norm is one and the normalization of a vector is that vector divided by its norm, and this is a unit vector that points in the same direction as the vector we are normalizing. We’ll see an application of this stuff. But first we’ll want to present a new Definition: Distance, which we will do in its own set of notes. Or I guess I should say in. It’s ownvideo!