"In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces."
A vector is a mathematical object that has both magnitude and direction. In linear algebra, vectors are usually represented as column matrices.
Introduction to Vectors: Basic definition of vectors, vector addition, and scalar multiplication operations.
Properties of Vectors: Commutative, associative, and distributive properties of vectors.
Vector Norm: Length or magnitude of a vector, Euclidean norm or magnitude of a vector in Euclidean space.
Vector Spaces: Vectors can form a linear space with certain properties. It can have a basis, dimension, and span.
Basis of Vectors: Every vector in a linear space can be written as a linear combination of a fixed set of vectors.
Coordinates: The numerical representation of a vector in a chosen basis with respect to a fixed origin.
Linear combinations: A linear combination of vectors is an expression in which the vectors are multiplied by constants and then added together.
Inner Product: The dot product of two vectors measure the angle between them and has properties like linearity, symmetry, and orthogonality.
Norms: Generalization of Euclidean norm: L1 norm, L2 norm, and P-norm.
Linear Independence: A set of vectors is linearly independent when no one vector can be expressed as a linear combination of the others.
Linear transformations: A function that maps vectors from one vector space to another and preserves linear combinations.
Matrices: Matrices can represent linear transformations and operations like addition, multiplication, and transposition.
Eigenvalues and Eigenvectors: Scalar values and vectors that do not change in direction under a linear transformation.
Diagonalization: Representing a matrix as a diagonal matrix through Eigenvalues and Eigenvectors decomposition.
Applications in Geometry: Vectors are used to represent points, lines, and planes in three-dimensional space.
Applications in Physics: Vectors are used to represent forces, velocities, accelerations, and other physical quantities.
Applications in Computer Science: Vectors are used to represent data structures, machine learning, and signal processing.
Tensor Products: Generalization of the outer product of two vectors to higher dimensional spaces.
Applications in other fields: Vectors are used in various fields like economics, chemistry, biology, and social sciences.
Row vector: A row vector is a sequence of numbers arranged horizontally. It is used to represent a single row of a matrix.
Column vector: A column vector is a sequence of numbers arranged vertically. It is used to represent a single column of a matrix.
Zero vector: A vector whose all components are zero is called the zero vector.
Position vector: A position vector is a vector that points from the origin to a given point in space.
Unit vector: A unit vector is a vector of length one. It is used to describe directions and orientation.
Normal vector: A normal vector is perpendicular to a given surface or object. It is commonly used in various math applications such as calculus, geometry, and physics.
Basis vector: A set of basis vectors span a vector space. Any vector in the space can be expressed as a linear combination of these basis vectors.
Eigen vector: An eigenvector is a non-zero vector that only changes its scale when it is multiplied by a given transformation matrix.
Projection vector: A projection vector is a vector that represents the projection of a given vector onto a given subspace.
Parallel vector: A vector that has the same direction as another vector but possibly a different magnitude.
Orthogonal vector: Two vectors are orthogonal if their dot product is zero. It means that the two vectors are perpendicular to each other.
Gradient vector: The gradient vector is a vector that points in the direction of the maximum rate of change of a given function.
Curl vector: The curl vector is a vector that represents the rotation of a vector field.
Divergence vector: The divergence vector is a vector that represents the change in the density of a vector field.
"Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Such quantities are represented by geometric vectors in the same way as distances, masses, and time are represented by real numbers."
"The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length."
"Both geometric vectors and tuples can be added and scaled."
"These vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces..."
"a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar)..."
"...quantities that have both a magnitude and a direction, such as displacements, forces, and velocity."
"Both geometric vectors and tuples can be added and scaled..."
"...a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors."
"A vector space formed by geometric vectors is called a Euclidean vector space."
"A vector space formed by tuples is called a coordinate vector space."
"...extension field, polynomial rings, algebras, and function spaces."
"The term vector is generally not used for elements of these vector spaces..."
"...and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces."