Linear Algebra

Linear Combination

An expression formed by multiplying a set of vectors by scalar coefficients and adding the results, written as $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n$.

Span

The set of all possible linear combinations of a given collection of vectors, which forms a subspace of the vector space.

Linear Independence

A property of a set of vectors where the equation $c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = \mathbf{0}$ is satisfied only when $c_1 = c_2 = \dots = c_k = 0$.

Basis

A set of vectors in a vector space that is both linearly independent and spans the entire space.

Dimension

The number of vectors in any basis for a given vector space.

Rank of a Matrix

The maximum number of linearly independent column (or row) vectors in a matrix, representing the dimension of its column space.

Nullity

The dimension of the null space (kernel) of a matrix $A$, representing the dimension of the solution space to $A \mathbf{x} = \mathbf{0}$.

Rank-Nullity Theorem

For an $m \times n$ matrix $A$, this theorem states that $\text{rank}(A) + \text{nullity}(A) = n$.

Determinant

A scalar value computed from a square matrix that indicates whether the matrix is invertible, where a value of $0$ means the matrix is singular.

Eigenvalue

A scalar $\lambda$ such that $A \mathbf{v} = \lambda \mathbf{v}$ for a square matrix $A$ and a non-zero vector $\mathbf{v}$.

Eigenvector

A non-zero vector $\mathbf{v}$ that, when multiplied by a square matrix $A$, results in a scalar multiple of itself according to $A \mathbf{v} = \lambda \mathbf{v}$.

Characteristic Equation

The equation $\det(A - \lambda I) = 0$ used to calculate the eigenvalues of a square matrix $A$.

Dot Product

An operation taking two vectors that yields a scalar given by $\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i = \|\mathbf{u}\| \|\mathbf{v}\| \cos(\theta)$.

Cross Product

A binary operation on two vectors in three-dimensional space producing a vector orthogonal to both, with magnitude equal to $\parallel\mathbf{u}\parallel \parallel\mathbf{v}\parallel \sin(\theta)$.

Orthogonal Projection

The projection of a vector $\mathbf{u}$ onto a non-zero vector $\mathbf{v}$, given by $\text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}$.