Linear Algebra Fundamentals

Vectors

A vector in $\mathbb{R}^n$ is an ordered $n$-tuple:

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

Dot Product

For vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$:

$$ \mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i = |\mathbf{u}| |\mathbf{v}| \cos \theta $$

Cross Product (in $\mathbb{R}^3$)

$$ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix} $$

Matrices

An $m \times n$ matrix:

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

Matrix Multiplication

If $A$ is $m \times p$ and $B$ is $p \times n$, then:

$$ (AB){ij} = \sum{k=1}^p a_{ik} b_{kj} $$

Determinant (2×2)

$$ \det \begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc $$

Eigenvalues and Eigenvectors

For a square matrix $A$, if:

$$ A\mathbf{v} = \lambda \mathbf{v} $$

then $\lambda$ is an eigenvalue and $\mathbf{v}$ is the corresponding eigenvector.

Characteristic Equation

$$ \det(A - \lambda I) = 0 $$

Applications

The singular value decomposition (SVD):

$$ A = U \Sigma V^T $$

is widely used in data science for dimensionality reduction (PCA).