Linear Algebra: Reference Tables and Quick Lookups

Abstract

This article compiles essential definitions, formulas, and properties from linear algebra into a structured reference guide. It covers matrix operations, determinants, eigenvalues, and diagonalization—topics central to undergraduate linear algebra courses. The guide prioritizes clarity and accessibility for students seeking quick verification of key concepts and computational procedures.

Background

Linear algebra provides the mathematical framework for understanding systems of linear equations, vector spaces, and linear transformations. Practitioners frequently need to reference formulas and properties without re-deriving them. This article organizes core material into digestible sections, each grounded in course notes and suitable for consultation during problem-solving or exam preparation.

Key Results

Matrix Multiplication

^{[matrix-multiplication]} defines the product of two matrices. For matrices $A$ (size $m \times n$) and $B$ (size $n \times p$), the entry in row $i$ and column $j$ of the product $AB$ is computed as:

$$(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$$

This operation is non-commutative: in general, $AB \neq BA$. Matrix multiplication is foundational for representing linear transformations and solving systems of equations.

Determinant: Definition and Computation

The determinant is a scalar assigned to a square matrix that encodes critical information about invertibility and volume scaling. ^{[determinant-of-a-matrix]} provides the explicit formula for $2 \times 2$ matrices:

$$\det(A) = ad - bc \quad \text{for} \quad A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

For larger matrices, determinants are computed recursively using minors and cofactors. A non-zero determinant indicates the matrix is invertible; a zero determinant signals linear dependence among rows or columns.

Determinant Properties

^{[determinant-properties]} and ^{[determinant-properties]} enumerate key properties:

1. Transpose invariance: $\det(A^T) = \det(A)$
2. Row swap: Swapping two rows multiplies the determinant by $-1$
3. Row scaling: Multiplying a row by scalar $c$ multiplies the determinant by $c$
4. Row addition: Adding a multiple of one row to another does not change the determinant
5. Triangular matrices: The determinant equals the product of diagonal entries

These properties enable efficient determinant computation via row reduction.

Eigenvalues and the Characteristic Equation

^{[eigenvalues-of-a-matrix]} defines eigenvalues as scalars $\lambda$ satisfying:

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

This characteristic equation is solved to find all eigenvalues of a square matrix $A$. Eigenvalues reveal how much corresponding eigenvectors are scaled under the transformation represented by $A$, and they appear in applications ranging from stability analysis to principal component analysis.

Column Space and Null Space

^{[basis-of-column-space]} states that the basis of the column space $\text{Col}(A)$ consists of the pivot columns in row echelon form. The dimension of the column space equals the number of pivot columns.

^{[basis-of-null-space]} defines the null space as the solution set to $Ax = 0$. Its basis is found by solving this homogeneous system, and its dimension equals the number of free variables. Together, these spaces characterize the range and kernel of the linear transformation defined by $A$.

Diagonalization

^{[diagonalizable-matrix]} and ^{[diagonalizable-matrix]} define a diagonalizable matrix as one expressible in the form:

$$A = PDP^{-1}$$

where $D$ is diagonal (containing eigenvalues) and $P$ is invertible (with eigenvectors as columns). Diagonalization simplifies matrix powers and systems of differential equations. A matrix is diagonalizable if and only if it possesses a complete set of linearly independent eigenvectors.

Matrix Equations and Inversion

^{[matrix-equation-solution]} addresses solving $B^{-1}(A - X) = AX$ for $X$, yielding:

$$X = (BA + I)^{-1}A$$

provided $BA + I$ is invertible. Similarly, ^{[matrix-inversion-formula]} solves $B^{-1}(A + X) = AX$ to obtain:

$$X = (BA - I)^{-1}A$$

These results illustrate systematic manipulation of matrix equations using properties of invertible matrices.

Worked Examples

Example 1: Computing a $2 \times 2$ Determinant

For $A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$:

$$\det(A) = (3)(4) - (2)(1) = 12 - 2 = 10$$

Since $\det(A) \neq 0$, the matrix is invertible.

Example 2: Row Operation and Determinant

If $B$ is obtained from $A$ by swapping two rows, then $\det(B) = -\det(A)$ ^{[determinant-properties]}. If $C$ is obtained by multiplying one row of $A$ by $5$, then $\det(C) = 5 \cdot \det(A)$.

Example 3: Identifying Pivot Columns

After row reduction, the columns containing leading entries (pivots) form a basis for the column space ^{[basis-of-column-space]}. The number of such columns is the rank and also the dimension of the column space.

References

• ^{[matrix-multiplication]}
• ^{[determinant-of-a-matrix]}
• ^{[determinant-properties]}
• ^{[determinant-properties]}
• ^{[determinant-of-a-matrix]}
• ^{[eigenvalues-of-a-matrix]}
• ^{[basis-of-column-space]}
• ^{[basis-of-null-space]}
• ^{[diagonalizable-matrix]}
• ^{[diagonalizable-matrix]}
• ^{[matrix-equation-solution]}
• ^{[matrix-inversion-formula]}

AI Disclosure

This article was drafted with the assistance of an AI language model. The mathematical content and structure are derived from the cited Zettelkasten notes; the AI was used to organize, paraphrase, and format the material for clarity and readability. All claims are traceable to the source notes listed in the References section.