Linear Algebra

Introduction to Linear Algebra

Linear algebra is a field of mathematics that focuses on studying vectors, matrices, and the relationships between them, forming the mathematical framework for analyzing structures and transformations in multidimensional spaces. It introduces powerful tools to understand and solve problems where quantities interact linearly, making it fundamental to numerous disciplines.

This field begins with vectors—quantities that have both magnitude and direction—and their operations, such as addition and scaling. It extends to matrices, which are grid-like arrangements of numbers used to represent systems of equations or transformations. Learning how to manipulate matrices and understand their properties is a key part of linear algebra.

Students also explore vector spaces, the environments in which vectors live, and subspaces, which reveal structure and constraints within these spaces. Concepts like linear independence, span, and basis give insight into how vectors relate and interact. The study of linear transformations, which describe how vectors change under operations like rotations or scaling, helps build a deeper understanding of the subject.

Eigenvalues and eigenvectors, pivotal concepts in linear algebra, allow students to uncover hidden properties of transformations. Techniques like solving systems of equations, matrix decomposition, and understanding projections or orthogonality are practical outcomes of this study.

Ultimately, linear algebra provides a foundation for solving abstract and applied problems, developing skills to think logically, recognize patterns, and simplify complex systems. It equips students with a versatile toolkit for further studies in mathematics, sciences, engineering, and beyond.

Linear Algebra Formulas

Explore Linear Algebra formulas with explanations and examples

Vector Addition

\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix}

Scalar Multiplication

c\mathbf{v} = \begin{bmatrix} c \cdot v_1 \\ c \cdot v_2 \\ \vdots \\ c \cdot v_n \end{bmatrix}

Dot Product (Inner Product)

\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \dots + u_nv_n

Cross Product

\mathbf{u} \times \mathbf{v} = \begin{bmatrix} u_2v_3 - u_3v_2 \\ u_3v_1 - u_1v_3 \\ u_1v_2 - u_2v_1 \end{bmatrix}

Matrix Addition

\mathbf{A} + \mathbf{B} = \begin{bmatrix} a_{11} + b_{11} & \dots & a_{1n} + b_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} + b_{m1} & \dots & a_{mn} + b_{mn} \end{bmatrix}

Scalar Multiplication of a Matrix

c\mathbf{A} = \begin{bmatrix} c \cdot a_{11} & \dots & c \cdot a_{1n} \\ \vdots & \ddots & \vdots \\ c \cdot a_{m1} & \dots & c \cdot a_{mn} \end{bmatrix}

Matrix Multiplication

(\mathbf{AB})_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}

Determinant of a 2x2 Matrix

\det(\mathbf{A}) = ad - bc

Inverse of a 2x2 Matrix

\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Eigenvalues and Eigenvectors

\mathbf{A}\mathbf{x} = \lambda\mathbf{x}

Cramer's Rule

x_i = \frac{\det(\mathbf{A}_i)}{\det(\mathbf{A})}

Rank of a Matrix

The rank is the maximum number of linearly independent rows or columns.

Linear Transformation

T(c\mathbf{u} + d\mathbf{v}) = cT(\mathbf{u}) + dT(\mathbf{v})

Orthogonality

\mathbf{u} \cdot \mathbf{v} = 0

Norm of a Vector

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2}

Unit Vector

\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}

Projection of a Vector

\text{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \right) \mathbf{v}