Section 1.1: What is a Matrix?

A matrix can be described as a rectangular arrangement of data. For instance, a numerical data table qualifies as a matrix. Imagine a simple matrix composed of three columns and three rows, holding information about age, annual income, and ID for three individuals. To represent this data in Python, we can utilize the Numpy library as follows:

import numpy as np

# Matrix A:

matrix_A = np.array([[23, 50600, 647],

[56, 76900, 254],

[44, 103400, 165]])

If we print Matrix A, we can visualize our dataset. A matrix is denoted by a capital letter and is characterized by 'n' (or 'i') rows and 'm' (or 'j') columns. The Numpy library is widely regarded for its capabilities in linear algebra and matrix manipulation.

For example, the shape of Matrix A can be determined with:

matrix_A.shape

This yields:

[OUT] (3, 3)

Matrix A is a rectangular matrix with a shape of 3x3. However, matrices can also have shapes such as 1x3 or 3x1:

matrix_B = np.array([[1, 5, 7]])

matrix_B.shape

[OUT] (1, 3)

matrix_C = np.array([[1], [5], [7]])

matrix_C.shape

[OUT] (3, 1)

Section 1.2: The Identity Matrix

The identity matrix is a square matrix of any dimension that features ones on its main diagonal, with all other elements being zeros:

identity_matrix = np.array([[1, 0, 0, 0],

[0, 1, 0, 0],

[0, 0, 1, 0],

[0, 0, 0, 1]])

print(identity_matrix)

Elementary Row Operations

Elementary row operations are a set of transformations used to manipulate the values within a row. There are three main types of these operations:

• Multiplying a row by a scalar
• Adding one row to another
• Swapping two rows

For instance, let's define a new matrix A:

A = np.array([[2, 4, 6], [8, 10, 12], [14, 16, 18]])

print(A)

To multiply a row by a scalar, simply multiply every element in that row. For example, multiplying the first row by 3 results in:

A_after_line_1_x3 = np.array([[(2*3), (4*3), (6*3)],

[8, 10, 12],

[14, 16, 18]])

print(A_after_line_1_x3)

In the next part of this series, we will learn how to perform matrix multiplication. You can also use the identity matrix for elementary row operations.

To create an identity matrix matching the size of matrix A:

A_identity = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 1]])

print(A_identity)

Next, we apply an elementary row operation to the identity matrix:

A_identity = np.array([[(3*1), (3*0), (3*0)],

[0, 1, 0],

[0, 0, 1]])

print(A_identity)

Now, we multiply the identity matrix by matrix A using Numpy's matmul function, keeping in mind the order of multiplication:

result = np.matmul(A_identity, A)

print(result)

Adding and Exchanging Rows

To add one row to another, you can perform:

A_add_row1_to_row2 = np.array([[2, 4, 6],

[(2 + 8), (10 + 4), (6 + 12)],

[14, 16, 18]])

print(A_add_row1_to_row2)

Using the identity matrix, the process is similar:

A_identity = np.array([[1, 0, 0],

[1, 1, 0],

[0, 0, 1]])

result = np.matmul(A_identity, A)

print(result)

To swap two rows in matrix A, you would do:

A_exchange_row2_row3 = np.array([[2, 4, 6],

[14, 16, 18],

[8, 10, 12]])

print(A_exchange_row2_row3)

Using the identity matrix, the exchange looks like this:

A_identity = np.array([[1, 0, 0],

[0, 0, 1],

[0, 1, 0]])

result = np.matmul(A_identity, A)

print(result)

Matrices in Linear Equations

Matrices can also represent systems of linear equations, which define relationships among variables. For example:

• x + y + 2z = 9
• 2x + 4y - 3z = 1
• 3x + 6y - 5z = 0

The leading coefficient is the first non-zero element in each row of a matrix. For instance, in the matrix below:

The leading coefficient in row 1 is 2, in row 2 is 14, and in row 3 is 8.

Gaussian Elimination and Row Reduction

Gaussian elimination is a method that uses elementary row operations to convert a matrix so that as many elements as possible in the lower left corner become zeros. When a matrix reaches this stage, it is in row echelon form. Continuing the process until all leading coefficients are one results in a reduced row echelon form.

Finding Solutions for Linear Equations

By applying elementary row operations, we can simplify a matrix to solve for the variables in a system of equations. For example, if we have:

A = np.array([[1, 1, 2],

[2, 4, -3],

[3, 6, -5]])

B = np.array([9, 1, 0])

We can find the solution using Numpy's linalg.solve function:

solution = np.linalg.solve(A, B)

print(solution)

Now it's your turn to explore and solve systems of linear equations, either manually or with Python!

References and Further Reading

• Elementary Linear Algebra, Applications Version, 9th Edition, Howard Anton and Chris Rorres.
• Maths for Computing and Information Technology, Frank Giannasi and Robert Low.

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