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qmlcourse/_toc.yml

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       title: 🟦 About quantum computers
     - file: book/intro/en/prerequisite.md
       title: ⬜ Prerequisite
+- part: Linear Algebra and Numpy
+  chapters:
+    - file: book/linalg/en/vectors.md
+      title: 🟦 Vectors

qmlcourse/book/linalg/en/vectors.md

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+(vectors)=
+# Vectors
+
+Author(s):
+
+- [El-Ayyass Dani](http://github.com/dayyass)
+
+Translation:
+
+- [Pronkin Alexey](http://github.com/alexey-pronkin)
+- [Zheltonozhskii Evgenii](https://github.com/Randl)
+- [Trohimenko Viktor](https://github.com/vtrokhymenko)
+
+
+## Definition
+
+The term `vector` has several interpretations: mathematical, geometric, physical, etc. The exact meaning of the term depends on the context.
+
+Formally, a vector is defined as an element of vector space -- a set with **addition** and **multiplication of a vector by a number (scalar)** operations, which satisfy [8 axioms](https://en.wikipedia.org/wiki/Vector_space).
+
+For simplicity of understanding, let us consider the rectangular (Cartesian) coordinate system in the plane, familiar from school days -- two perpendicular to each other axes $x$ and $y$, selected unit vectors (orthogonal) $\mathbf{e}_1$, $\mathbf{e}_2$ on them and the origin of coordinates.
+
+The vector $\mathbf{a}$ in this coordinate system can be written as follows: $\mathbf{a} = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$
+
+**Note 1**: Vector coordinates do not uniquely define its position in the plane, but only the position of the vector's end relative to its start. For example, a vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ can be drawn either from the origin $\begin{pmatrix} 0 , 0 \end{pmatrix}$ with the end at $\begin{pmatrix} 3 , 4 \end{pmatrix}$, or from an arbitrary point, e.g. $\begin{pmatrix} 1 , 1 \end{pmatrix}$ with the end at $\begin{pmatrix} 4 , 5 \end{pmatrix}$. Both of these vectors correspond to 3 orthogonal unite vectors on the $x$-axis and 4 on the $y$-axis. Normally, unless otherwise stated, it is assumed that the vector is deferred from the origin.
+
+**Note 2**: A vector can be represented either as a column vector $\begin{pmatrix} a_1 \ a_2 \end{pmatrix}$ or as a row vector $\begin{pmatrix} a_1 \ a_2 \end{pmatrix}$. Hereinafter, by a vector we mean a column vector, unless otherwise stated.
+
+The notion of a vector in the plane can be generalized to 3-dimensional space, and, in general, to $n$-dimensional space (which can no longer be visualized):
+
+$$
+    \mathbf{a} = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + ... + a_n \mathbf{e}_n = \begin{pmatrix} a_1 \ a_2 \\ ... \ a_n \end{pmatrix}
+$$
+
+## Vector operations
+
+As stated earlier, there are two basic operations on vectors in the formal definition:
+
+- addition:
+
+$$
+    \mathbf{a} + \mathbf{b} = (a_1 + b_1) \mathbf{e}_1 + (a_2 + b_2) \mathbf{e}_2 + ... + (a_n + b_n) \mathbf{e}_n = \ = \begin{pmatrix} a_1 \ a_2 \\ ... \ a_n \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ ... \ b_n \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \ a_2 + b_2 \\ ... \ a_n + b_n \end{pmatrix}
+$$
+
+- multiplication of a vector by a number (scalar):
+
+$$
+    \lambda \mathbf{a} = \lambda a_1 \mathbf{e}_1 + \lambda a_2 \mathbf{e}_2 + ... + \lambda a_n \mathbf{e}_n = \lambda \begin{pmatrix} a_1 \ a_2 \\ ... \ a_n \end{pmatrix} = \begin{pmatrix} \lambda a_1 \\\ \lambda a_2 \\ ... \ \lambda a_n \end{pmatrix}
+$$
+
+The operation of multiplying a vector by a number can be interpreted geometrically as a contraction/extension of a vector.
+
+Using these two operations, we can define **linear combinations of vectors**: $\alpha_1 \mathbf{a}_1 + \alpha_2 \mathbf{a}_2 + ... + \alpha_n \mathbf{a}_n ,$ where $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_n$ are vectors and $\alpha_1, \alpha_2, ..., \alpha_n$ are numbers.
+
+## The norm (length) of a vector
+
+In linear algebra, the term **norm** is used to generalize the notion of vector length. It can be said that the notions of length and norm are equivalent.
+
+Formally the norm is defined as a functional in the vector space satisfying [3 axioms](https://en.wikipedia.org/wiki/Norm_(mathematics)), norm is mapping elements of this space (vectors) to the set of non-negative real numbers.
+
+There are many functionals satisfying this definition of norm, but we will consider the most commonly used one -- **Euclidean norm**.
+
+For simplicity, let us consider a vector in the plane. Geometrically, it is a directional segment. The directionality of a vector has no effect on its length, so we can work with it as a segment in the plane and calculate its length using school formulas.
+Note that the vector coordinates correspond to the numbers multiplying coordinate axis orthogonal vectors, so the vector length formula is: $\lVert \mathbf{a} \rVert = \sqrt{a_1^2 + a_2^2}$. Accordingly, in general, the formula is  $\lVert \mathbf{a} \rVert = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$.
+
+## Scalar product
+
+One of the most common operations on two vectors is the so-called **scalar product**, the result of which is a number (scalar) -- hence the name of the operation.
+
+**Note**: In addition to scalar product there is also [vector product or cross product](https://en.wikipedia.org/wiki/Cross_product) over a pair of vectors, the result of which is a vector. There is also [triple product](https://en.wikipedia.org/wiki/Triple_product) over a triplet of vectors, the result of which is a number. These operations will not be covered in this course.
+
+Scalar product is used in determining the length of vectors and the angle between them. This operation has two definitions:
+
+- Algebraic: $ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + ... + a_n b_n$
+- geometric: $\mathbf{a} \cdot \mathbf{b} = \lVert \mathbf{a} \rVert \lVert \mathbf{b} \rVert \cos{\\theta},$ where $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$.
+
+Using both of these definitions, we can derive a formula for calculating the cosine of the angle between the vectors:
+
+$$
+    \cos{\theta} = \frac{\mathbf{a} \cdot \mathbf{b}}{\lVert \mathbf{a} \rVert \lVert \mathbf{b} \rVert} = \frac{a_1 b_1 + a_2 b_2 + ... + a_n b_n}{\sqrt{a_1^2 + a_2^2 + ... + a_n^2} \sqrt{b_1^2 + b_2^2 + ... + b_n^2}}
+$$
+
+Using this formula we can arrive at one of the main properties of scalar product, which is that **two vectors are perpendicular if and only if their scalar product is 0**: $\mathbf{a} \perp \mathbf{b} \leftrightarrow \cos{\theta} = 0 \leftrightarrow \mathbf{a} \cdot \mathbf{b} = 0$
+
+The scalar product can be used to calculate the vector norm as follows: $\lVert \mathbf{a} \rVert = \sqrt{\mathbf{a} \cdot \mathbf{a}}$
+
+## Linear independence
+
+One of the fundamental concepts of linear algebra is **linear dependence/independence**.
+
+To define this concept, consider a set of several vectors. A set of vectors is **linearly dependent** if there exists nontrivial linear combination of vectors of this set (at least one element of this combination is not 0) equal to a zero vector (a vector consisting only of 0):
+
+$$
+    \lambda_1 \mathbf{a}_1 + \lambda_2 \mathbf{a}_2 + ... + \lambda_m \mathbf{a}_m = \begin{pmatrix} 0 \\ 0 \\ ... \\ 0 \end{pmatrix}, \ \exists \lambda_1 \not = {0} \vee \lambda_2 \not = {0} \vee ... \vee \lambda_m \not = {0}
+$$
+
+If a set of vectors is not linearly dependent, that is, there is no nontrivial linear combination of vectors of a given set equal to the zero vector, then such a set of vectors is called **linearly independent**.
+
+**Example**:
+
+- A linearly independent set of vectors (only the zero coefficients of a linear combination lead to a zero vector):
+
+$$
+    \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\\ 0 \\ 1 \end{pmatrix} \rightarrow 0 \cdot \begin{pmatrix} 1 \\ 0 \\ 0  \end{pmatrix} + 0 \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + 0 \cdot \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
+$$
+
+- linearly dependent set of vectors (there are non-zero coefficients of the linear combination that lead to the zero vector):
+
+$$
+    \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} \rightarrow 2 \cdot \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + 3 \cdot \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} - 1 \cdot\begin{pmatrix} 2 \\ 3 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}
+$$
+
+The following property can be deduced from the definition of linear dependence:
+A set of vectors is linearly dependent if and only if one of the elements of that set can be expressed through the remaining elements.
+
+**Note**: If vectors represent some characteristics of objects, then linear dependence can be interpreted as data redundancy.
+
+The notion of linear independence introduces the concept of **dimensionality** of a vector space -- the maximum number of linearly independent vectors in it.
+
+## Basis
+
+In the formal definition of a vector there are no quantitative interpretations of it, just two operations on vectors and 8 axioms.
+
+So where do quantitative measurements come from? To make this clear, we need to introduce the concept of **basis**.
+
+A basis is a finite set of vectors in a vector space such that any vector in that space can be uniquely represented as a linear combination of vectors of that set.
+
+Recall one of the examples above, where we represented a rectangular coordinate system in the plane and unit vectors $\mathbf{e}_1$, $\mathbf{e}_2$. $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$.
+In this example, we decompose the arbitrary vector $\mathbf{a}$ as follows:
+
+$$
+    \mathbf{a} = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 = a_1 \begin{pmatrix} 1 \\ 0 \end{pmatrix} + a_2 \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} a_1 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ a_2 \end{pmatrix} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}
+$$
+
+It turns out that the orthogonal vectors $\mathbf{e}_1$, $\mathbf{e}_2$ **are the basis of the two-dimensional vector space**, and by linear combination of these vectors we can uniquely represent any vector in this space.
+
+The question arises, is this basis unique in two-dimensional space, or not?
+
+The answer to this question is no. In fact, one can take any two vectors (**almost any**) and they will also form a basis, provided that any vector can be decomposed by their linear combination.
+
+**Example**.
+
+Suppose we have a vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ in a basis of unit orthogonal vectors, and we want to decompose it in another basis $\begin{pmatrix} -2 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ -3 \end{pmatrix}$:
+
+$$
+    \alpha_1 \begin{pmatrix} -2 \\ 0 \end{pmatrix} + \alpha_2 \begin{pmatrix} 0 \\ -3 \end{pmatrix} = \begin{pmatrix} 2 \\3 \end{pmatrix}
+$$
+
+From where we can find that $\alpha_1 = -1$, $\alpha_2 = -1$:
+
+$$
+    -1 \begin{pmatrix} -2 \\ 0 \end{pmatrix} + -1 \begin{pmatrix} 0 \\ -3 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}
+$$
+
+So, vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ in the basis of unit orthogonal vectors is represented as $\begin{pmatrix} -1 \\ -1 \end{pmatrix}$ in the $\begin{pmatrix} -2 \\ 0 \end{pmatrix} \begin{pmatrix} 0 \\ -3 \end{pmatrix}$.
+
+But, as mentioned before, **not any set of vectors**  makes a basis.
+
+For example, through the set of vectors $\begin{pmatrix} -2 \\ 0 \end{pmatrix}$, $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$ cannot decompose the vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, so this set of vectors is not a basis.
+
+What is the fundamental difference between these bases, and can the basis of a two-dimensional space consist of more or fewer vectors than 2, for example?
+
+Linear algebra has an answer to this:
+
+**Any $n$ linearly independent vectors of $n$-dimensional vector space form the basis of that space.**
+
+Exactly due to the linear dependence, the vectors $\begin{pmatrix} -2 \\ 0 \end{pmatrix}$, $\begin{pmatrix} -3 \\ 0 \end{pmatrix}$ can't be the basis of a two-dimensional space.
+
+## What did we learn?
+
+- The definition of a vector;
+- The operations on vectors;
+- The vector norm (length);
+- The scalar product;
+- The linear independence;
+- The basis.