Linear Algebra

TL;DR

• Provides methods to represent and manipulate data using vectors and matrices.
• Core operations include vector addition and matrix multiplication.
• Widely applied in physics, engineering, and computer science for tasks such as forces, spatial transformations, algorithms, and machine learning.

Definition

Linear algebra is a branch of mathematics that deals with the study of linear equations, vectors, and matrices.

Explanation

Linear algebra studies mathematical objects such as vectors and matrices and the operations that combine them. Vectors are objects with magnitude and direction and can be represented by components (for example, displacements in x and y). Vector addition combines two or more vectors by adding their corresponding components. Matrices are rectangular arrays of numbers, and matrix multiplication combines two or more matrices by multiplying their corresponding elements. Linear algebra provides concise, systematic tools for representing and manipulating data, enabling applications across multiple fields.

Examples

Vector addition

Given two vectors A and B with components (3, 4) and (2, 5), their sum C is:

$$A = (3, 4),\quad B = (2, 5),\quad C = (3+2,\, 4+5) = (5, 9)$$

Matrix multiplication

Given matrices A and B with elements (1, 2, 3) and (4, 5, 6), respectively, the product is presented as:

$$A = (1, 2, 3),\quad B = (4, 5, 6),\quad C = (14 + 25 + 36,\, 14 + 25 + 36,\, 14 + 25 + 3*6) = (32, 32, 32)$$

Use cases

• Physics: compute net forces or represent transformations in space.
• Engineering: design algorithms for solving optimization problems.
• Computer science: perform machine learning tasks such as classification or regression.

Related terms

• Vectors
• Vector addition
• Matrices
• Matrix multiplication
• Linear equations