Mathematical formulas

Algebra

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1Numbers and algebraic properties
2Powers, exponents, and radicals
3Algebraic expressions
4Notable products and identities
5Factorization
6Rational expressions and partial fractions
7Equations
8Inequalities
9Systems of equations
10Functions
11Polynomials
12Exponentials and logarithms
13Complex numbers
14Sequences, finite series, and recurrences
15Vectors
16Matrices
17Determinants and inverses
18Vector spaces
19Linear transformations
20Eigenvalues and diagonalization
21Orthogonality and projections
22Least squares
23Matrix decompositions
24Matrix norms and conditioning
25Boolean algebra
26Modular arithmetic
27Algebraic structures
28Linear codes and error correction
29Relationship maps

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Algebra
Systems of equations

Section 9

Systems of equations

2×2 linear system

fundamentalviz

\begin{cases}a_1x+b_1y=c_1\\a_2x+b_2y=c_2\end{cases}

Two linear equations in two unknowns.

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intermedioviz

A\mathbf x=\mathbf b

Represents a linear system in matrix notation.

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Augmented matrix

intermedioviz

[A\mid\mathbf b]

Collects coefficients and constant terms.

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Elementary row operations

intermedioviz

R_i\leftrightarrow R_j;\;R_i\leftarrow cR_i;\;R_i\leftarrow R_i+cR_j

Foundation of Gaussian and Gauss–Jordan elimination.

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Rouché–Capelli theorem

intermedioviz

\operatorname{rank}(A)=\operatorname{rank}([A|b])=n \quad\text{or}\quad \operatorname{rank}(A)=\operatorname{rank}([A|b])<n \quad\text{or}\quad \operatorname{rank}(A)\neq\operatorname{rank}([A|b])

Classifies systems according to ranks and the number of unknowns.

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