Introduction to Vectors and Vector Operations

• Vectors, matrices, and systems of linear equations are fundamental concepts.
• These are covered in linear algebra.

Vectors

• A vector has a direction, a sense, and a magnitude.

Vector Operations

• Vectors can be added together: $\overrightarrow{u} + \overrightarrow{v} = \overrightarrow{w}$
• Vectors can be multiplied by a scalar: $\lambda \overrightarrow{u} = \overrightarrow{w}$
• The dot product is defined as: $\overrightarrow{u} \cdot \overrightarrow{v} = ||\overrightarrow{u}|| \cdot ||\overrightarrow{v}|| \cdot \cos(\theta)$
• $\theta$ is the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$.
• The cross product (in 3D) is defined as: $\overrightarrow{u} \times \overrightarrow{v} = \overrightarrow{w}$
• $\overrightarrow{w}$ is orthogonal to $\overrightarrow{u}$ and $\overrightarrow{v}$.
• The magnitude of $\overrightarrow{w}$ is $||\overrightarrow{w}|| = ||\overrightarrow{u}|| \cdot ||\overrightarrow{v}|| \cdot \sin(\theta)$.
• In n-dimensional space, a vector is represented by an n-tuple of real numbers: $\overrightarrow{u} = (u_1, u_2,..., u_n)$

Matrices

• A matrix is a table of numbers with m rows and n columns and dimension m x n.

Matrix Operations

• $A + B = C$, where $c_{ij} = a_{ij} + b_{ij}$ (A and B must have the same dimension)
• $\lambda A = C$, where $c_{ij} = \lambda a_{ij}$
• Matrix multiplication: $A \cdot B = C$, where $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$
• The number of columns of A must equal the number of rows of B for matrix multiplication.
• Transposition: $A^T$, where $(a^T){ij} = a{ji}$

Matrix Types

• Square matrix: m = n
• Identity matrix: A square matrix with 1s on the diagonal and 0s elsewhere.
• Diagonal matrix: A square matrix with non-zero elements only on the diagonal.
• Symmetric matrix: $A = A^T$
• Orthogonal matrix: $A^T = A^{-1}$

Determinant

• The determinant of a square matrix A, denoted det(A) or |A|, is a scalar.
• For a 2x2 matrix, $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $det(A) = ad - bc$

Inverse

• The inverse of a square matrix A, denoted $A^{-1}$, satisfies $A \cdot A^{-1} = A^{-1} \cdot A = I$, where I is the identity matrix.

Systems of Linear Equations

• A system of linear equations can be represented as $Ax = b$ where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.

Solving Systems of Linear Equations

• Gaussian elimination transforms the system into an equivalent, easier-to-solve system using elementary row operations on the augmented matrix [A | b].
• Cramer's rule states that if det(A) $\neq$ 0, the system has a unique solution given by $x_i = \frac{det(A_i)}{det(A)}$.
• $A_i$ is the matrix obtained by replacing the i-th column of A with the vector b.

Eigenvalues and eigenvectors

• For a square matrix A, an eigenvector $\overrightarrow{v}$ is a non-zero vector such that $A\overrightarrow{v} = \lambda \overrightarrow{v}$, where $\lambda$ is an eigenvalue of A.
• Eigenvalues are found by solving the characteristic equation: $det(A - \lambda I) = 0$, where I is the identity matrix.