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
Vectors

Section 15

Vectors

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\mathbf v=(v_1,\ldots,v_n)^T

Fundamental object of linear algebra.

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\|\mathbf v\|_2=\sqrt{\sum_i v_i^2}

Euclidean length of a vector.

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\hat{\mathbf v}=\frac{\mathbf v}{\|\mathbf v\|}

Normalizes a nonzero vector.

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\mathbf u\cdot\mathbf v=\sum_i u_iv_i=\|u\|\|v\|\cos\theta

Measures alignment and allows defining angles and orthogonality.

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Angle between vectors

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\cos\theta=\frac{u\cdot v}{\|u\|\|v\|}

Obtains the angle between two nonzero vectors.

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Euclidean distance

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d(u,v)=\|u-v\|_2

Distance between two points/vectors.

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Linear combination

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w=c_1v_1+\cdots+c_kv_k

Builds new vectors from a given set.

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