Introduction to Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear maps, and their representations in vector spaces and through matrices.

What are Vectors?

A vector is a mathematical object that has both magnitude and direction. In the context of linear algebra, vectors are often represented as ordered lists of numbers.

$$\vec{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix}$$

Vector Operations

Addition: Two vectors can be added component-wise:

$$\vec{a} + \vec{b} = \begin{pmatrix} a_1 + b_1 \ a_2 + b_2 \end{pmatrix}$$

Scalar Multiplication: A vector can be multiplied by a scalar:

$$c\vec{v} = \begin{pmatrix} cv_1 \ cv_2 \end{pmatrix}$$

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns.

$$A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}$$

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. For matrices $A$ (m×n) and $B$ (n×p), the product $C = AB$ is an m×p matrix where:

$$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

Applications

Linear algebra is used extensively in:

• Computer graphics and game development
• Machine learning and data science
• Physics and engineering
• Economics and optimization

Conclusion

Understanding vectors and matrices is essential for anyone working in mathematics, computer science, or related fields. These concepts form the foundation for more advanced topics like eigenvalues, linear transformations, and optimization algorithms.