## Numerical operations on vectors
In Python, numerical operations on vectors can be performed using various libraries such as NumPy. NumPy provides efficient and convenient functions for vectorized operations, making it easy to perform calculations on vectors.

Here are some common numerical operations on vectors, along with examples in Python:
### Addition
Adding two vectors together element-wise.

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
c = a + b
print(c)

### Subtraction
 Subtracting one vector from another element-wise.

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
c = a - b
print(c)

### Scalar Multiplication 
Multiplying a vector by a scalar value.

import numpy as np

a = np.array([1, 2, 3])
b = 2
c = a * b
print(c)

### Dot Product
 Computing the dot product between two vectors.   
The dot product of two vectors is a mathematical operation that yields a scalar value.
The dot product of two vectors `u` and `v` is calculated by taking the sum of the products of their corresponding components:

`u · v = u₁ * v₁ + u₂ * v₂ + u₃ * v₃ + ... + uₙ * vₙ`

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
c = np.dot(a, b)

# Dot Product = 1*4 + 2*5 + 3*6
print(c)

## Operations on matrices
Now let's consider numerical operations on **matrices**. 

### Addition and subtraction 
Matrices and vectors of the same shape can be added or subtracted element-wise.

import numpy as np

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

S = A + B
D = A - B
print(S)
print(D)

### Scalar Multiplication and Division
Each element of a matrix or vector can be multiplied or divided by a scalar value.
Example in Python:

import numpy as np

A = np.array([[1, 2], [3, 4]])
scalar = 2

B = scalar * A  # Scalar multiplication
C = A / scalar  # Scalar division

print(B)
print(C)


### Matrix Multiplication
Multiplication of two matrices is a binary operation combining two matrices to produce a new matrix.    To compute the result of the multiplication of two matrices, **the number of columns in the first matrix must equal the number of rows in the second matrix**.    
The dot product of matrices is calculated by taking the dot product of corresponding rows from the first matrix and columns from the second matrix.   
The resulting matrix will **have the same number of rows as the first matrix and the same number of columns as the second matrix**.
The process of matrix multiplication can be visualized as follows: 

import numpy as np

# Define matrices A and B
A = np.array([[1, 2, 3], [4, 5, 6]])  # Matrix A with shape (2, 3)
B = np.array([[7, 8], [9, 10], [11, 12]])  # Matrix B with shape (3, 2)

# Perform dot product of matrices A and B
C = np.dot(A, B)  # Resulting matrix C with shape (2, 2)

# Print the resulting matrix C
print("Resulting matrix C (A dot B):")
print(C)

# Element-wise calculation of C[0, 0]
# c11 = 1*7 + 2*9 + 3*11
print("C[0, 0] = 1*7 + 2*9 + 3*11 =", C[0, 0])

# Element-wise calculation of C[0, 1]
# c12 = 1*8 + 2*10 + 3*12
print("C[0, 1] = 1*8 + 2*10 + 3*12 =", C[0, 1])

# Element-wise calculation of C[1, 0]
# c21 = 4*7 + 5*9 + 6*11
print("C[1, 0] = 4*7 + 5*9 + 6*11 =", C[1, 0])

# Element-wise calculation of C[1, 1]
# c22 = 4*8 + 5*10 + 6*12
print("C[1, 1] = 4*8 + 5*10 + 6*12 =", C[1, 1])

>Note
>
>Pay attention that matrix multiplication operation is not commutative.  In general case, `A * B != B * A`.

What will be the result of multiplication matrix with shape `(4, 5)` by matrix with shape `(3, 4)`?

To study many applied disciplines, it is necessary to have basic knowledge of higher mathematics and linear algebra. Mathematics is often used in data analysis, system modeling, building machine learning models, etc. Let's look at some of the most important topics and learn how to use the mathematical apparatus to solve real problems.

Let's start with some basic definitions and concepts we'll use later. Consider the idea of a function, a numerical sequence, and its sum, and also understand what a coordinate system's basis is.

The simplest and most commonly used type of relationship is the linear relationship. Linear algebra is a branch of higher mathematics entirely devoted to linear functions and linear spaces. Let's look at some of the most important topics in linear algebra: vectors, matrices, solving linear equations, and solving the spectral problem for matrices.

Mathematical analysis is a discipline that allows you to analyze functions according to various criteria. Consider how to check numerical sequences for convergence, find the maximum/minimum values of functions, solve nonlinear equations, and use integrals to solve applied problems.

Mathematics for Data Analysis and Modeling

Numerical Operations On Vectors and Matrices

Numerical operations on vectors

Addition

Subtraction

Scalar Multiplication

Dot Product

Operations on matrices

Addition and subtraction

Scalar Multiplication and Division

Matrix Multiplication

Example

What will be the result of multiplication matrix with shape `(4, 5)` by matrix with shape `(3, 4)`?

Mathematics for Data Analysis and Modeling

Numerical Operations On Vectors and Matrices

Numerical operations on vectors

Addition

Subtraction

Scalar Multiplication

Dot Product

Operations on matrices

Addition and subtraction

Scalar Multiplication and Division

Matrix Multiplication

Example

What will be the result of multiplication matrix with shape (4, 5) by matrix with shape (3, 4)?

What will be the result of multiplication matrix with shape `(4, 5)` by matrix with shape `(3, 4)`?