Lecture 1. Vector space
Vector space
Our lecture is devoted to the examples of sets that have the same type of structure and common properties. These examples will allow us to introduce significant concepts of the vector space. So, let's start. First, we are going to consider the sets, we have already discussed, while solving systems of linear equations.
Let's take a zero equation, all the coefficients of which are equal to zero. The solution of this equation is any set of numbers, so we can write down the set of its solutions in this form.
That is, we have an arbitrary set of numbers R1, R2, and so on Rn, where Ri takes an arbitrary value.
The letter R with the upper index n indicates such a set.
Note that we fix the number n, that is, we take an arbitrary natural n and fix it.
We get such a set. The set of all ordered n-s.
The elements of this set are also called n-dimensional lines, n-dimensional vectors, or just vectors.
Please note that the term vector is used due to the fact that this object has a lot in common with ordinary geometric vectors.
We will talk about this soon.
As I have already mentioned, elements from the set Rn are called vectors, and the numbers Ri are called scalars.
Now let's look at some models in which the concept of ordered n-s arises.
First, we note that each equation can correlate with an n-set of its solutions.
This set can be written down in the following way. However, if the equation have no solutions, then it corresponds to an empty set.
If there is a solution, then we can compare every equation to the set of its solutions - the set of n-s.
Second, each equation correlates with a line of its coefficients, that is, a line of its matrix.
Next, let's look at some geometric examples.
Let's fix a certain coordinate system on the plane.
Then, we get that each point on the plane is uniquely determined by its coordinates, that is, by a pair of numbers.
In our example, the point has coordinates (2; 1).
Thus, a pair of numbers can identify a point on the plane in the same way.
A point in the geometric space is uniquely defined by a triplet of numbers, provided that we fix some coordinate system.
In this example, point A corresponds to a triplet of numbers (2;1;3).
Now let's revise the concept of a geometric vector. This concept was studied in a school mathematics course, and it is used in both mathematics and physics. Thus, the vector in geometry is an object that is defined by the direction and length.
We know that the vector can be represented as a directed segment.
Let's look at some examples of vectors. We indicate by E2 the set of all vectors parallel to one plane.
We can assume that all vectors lie in the same plane if we set them from points that lie in this plane.
If we introduce a coordinate system on the plane and set the vector from the coordinate origin, then the coordinates of the vector are the coordinates of its end.
Thus, for each vector, we match a pair of numbers, that is, a pair of its coordinates.
Similarly, we consider the set E3 - the set of all vectors of the geometric space.
In this case, each vector correlates with three numbers, which we also call coordinates.
Note that in general, when we consider an arbitrary n-dimensional vector, its elements are also called coordinates by analogy with a plane and space.
Now let's look at an example of a physical phenomenon.
You know that light is electromagnetic radiation emitted, for example, by a heated body or substance.
There is the following dependence on the radiation wavelength. Radiation and its intensity will determine the color as a class of all light emissions that are equally perceived by the human eye.
Let’s consider the set of all colors. We indicate this set by C3. The RGB color model is known; it is based on the fact that each color is a combination of three basic colors: red with the intensity R, green-G, and blue-B.
It is because there are only three colors that we use the number 3 in the notation. Thus, any color corresponds to a triple of RGB intensity.
Let's look at a specific example. The color indicated on the slide is light green. We can get this color by combining the base colors as follows - 200 R+240 G+ 120 B.
Thus, the initially taken color gives us a triple of numbers 200 240 120.
Now let's consider operations on ordered n-s of numbers. n-dimensional vectors can be added. To add them you need to add the corresponding coordinates of the given n-s.
You can also multiply n-s by a number. In this case, each coordinate is multiplied by the given number k.
After we have entered the operations, let's look at how these operations will be reflected in the examples discussed earlier.
While converting equations, we performed these transformations with the lines of the matrix.
That is, if we have two equations to be added, we add the left and right parts together.
Thus, we obtain the following equation. At the same time, after we place the variable out of brackets, the coefficients obtained are the sums of the originally given coefficients.
Thus, it all comes down to adding the lines of the corresponding matrix. That is, we take a line with the first coefficients, a line with the second coefficients and add these lines according to the rule presented earlier.
The same situation with multiplication: if we multiply both parts of the equation by the number r, it means that we multiply the n-s of coefficients by this number.
We can deal with geometric vectors. The operations with them also correspond to operations with their coordinates.
Initially, we add vectors either by the triangle rule or by the parallelogram rule.
By taking two vectors in some coordinate system and setting them from the coordinate origin, we can find their sum.
On the other hand, if we deal with the coordinates of these vectors, then in order to find the coordinates of the sum, we need to add up the original coordinates of these vectors.
So, the addition of vectors corresponds to the addition operation over their coordinates similarly to the two multiplication operations. If we take, for example, the vector A with coordinates and we multiply it by ½ we will get a vector whose length is two times less than the length of the original vector. In this case, if the number is greater than 0, the vector is co-directed with the original one.
If we multiply it by a number less than 0, then the vector will change its direction. For this example, we keep the direction and reduce the length by half.
In coordinate language, it means that each coordinate is multiplied by½.
Thus, we get a vector with coordinates (1;1/2) from the vector (2;1).
We can interpret operations with colors in our set C3 in the same way.
In this case, the addition of colors corresponds to the mixing of radiation, and the multiplication of colors by this number is understood as changes in the intensity of radiation. Thus, operations with colors are determined by operations with their codes. For example, we take a color with the code (0;100;50) and add it with the color that has the code (100;50;150) and we get the color close to blue, which has the code (100;200;200). As you can see, we add the corresponding coordinates similarly to the addition of triplets of numbers.
Similarly, the example with multiplication, if we take a brown color and multiply it by 2, we get another color, which is obtained by multiplying the source color code (100;50;50) by 2, thus we get the code (200;100;100).
Let's study the properties that the considered operations obey in the set of n-dimensional lines.
When formulating the following properties, we will use the following notation - r1 and r2 are scalars, and, that is, the numbers a, b, c are n-dimensional lines.
Evidently, when adding lines, we do not care about the order in which we perform this operation, that is, changing the places of the summands will not change the sum. In mathematics, they say that the addition operation is commutative. Second, if we consider the sum of the three summands, this sum does not depend on how we place the brackets. The addition operation is said to be associative.
Third, there is a zero vector, i.e. a vector made up of zero coordinates, that is, a line consisting of just zeros.
This vector is characterized by these equations. That is, adding a null vector to any vector a does not change the vector a.
Next, each vector a has an opposite vector indicated as -a.
In this case, the sum of the opposite vectors gives a zero vector.
The next property is that by multiplying the vector by one, we will not change the vector. The following properties are applied. Note that here a is a vector, r1 and r2 are scalars. Now property seven. It says that you can remove parentheses if we convert this equation from left to right, or vice versa, if we go from the right side of the equation to the left, we can take the common multiplier out of parentheses.
Similarly, r1 is a scalar, a and b are vectors.
If we want to multiply a scalar by the sum of vectors, we can multiply the scalar by each vector separately, and then add them.
Here are the main properties that the vectors under consideration obey. At the same time, note that it satisfies these properties, not only vectors of the line, but also, geometric vectors, which are directed segments.
It does not matter whether they are on a plane or in space.
So, any set on which the operations of adding vectors and multiplying a vector by a number are specified, if the specified property is fulfilled, is called a vector space.
Let's give a clear definition.
A non-empty set v, on which addition and multiplication by a number satisfying the conditions are specified, is called a vector space.
Our example, the set of n-dimensional lines, as we have already said, is a vector space.
It is an arithmetic n-dimensional space
Similarly, the sets E2 and E3 are also vector spaces.
Note that the zero vector here is a vector whose beginning and end coincide.
Thus, it's just a dot.
An example of the set C3, the set of all colors, can also become a vector space, but for this, you need to add the so-called abstract non-existent colors to the real colors.
Thus, we can say that all the examples considered, where we introduced the operations of addition and multiplication by a number, are vector spaces.