# Vector spaces

Put content here**Vector spaces** are one of the foundational structures in linear algebra. A vector space is a set of objects called *vectors* that can be added together and multiplied by scalars (numbers from a field, typically \(\mathbb{R}\) or \(\mathbb{C}\)), satisfying a specific set of axioms.
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## Overview
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Vector spaces generalize the familiar notion of Euclidean space \(\mathbb{R}^n\) to more abstract settings. The key insight is that many mathematical objects --- tuples of numbers, matrices, polynomials, functions, sequences --- can all be treated as vectors once we define appropriate addition and scalar multiplication operations.
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## Key Concepts in This Section
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- **Coordinate vector spaces** (\(\mathbb{R}^n\), \(\mathbb{C}^n\)): The prototypical examples where vectors are ordered tuples of numbers.
- **Axioms of a vector space**: The ten properties that any vector space must satisfy.
- **Linear combinations, spans, and subspaces**: Tools for building and analyzing subsets of vector spaces.
- **Linear independence and bases**: Understanding minimal generating sets and the notion of dimension.
- **Linear transformations**: Structure-preserving maps between vector spaces.
- **Orthogonality and projection**: Geometric concepts that extend to inner product spaces.
- **Abstract vector spaces**: Examples beyond coordinate spaces, including function spaces, polynomial spaces, and matrix spaces.
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Vector spaces provide the language for discussing systems of linear equations, linear transformations, eigenvalues, and much of modern mathematics and its applications in physics, engineering, and computer science.

# Parents

* Linear algebra