This section demonstrates that many different types of mathematical objects satisfy the vector space axioms. Each example below shows a specific set with defined addition and scalar multiplication operations that make it a vector space.
The diversity of examples illustrates the power of the abstract vector space concept: once a theorem is proved for general vector spaces, it automatically applies to coordinate spaces, function spaces, polynomial spaces, matrix spaces, and more.
Each node in this section verifies the vector space axioms for a particular mathematical structure.
Parents
Children
- The set containing only 0 is a vector space.Theorem: The set \(\{\mathbf{0}\}\) containing only the zero vector is a vector space over any field \(\mathbb{F}\)....
- R^n is a vector space.Theorem: \(\mathbb{R}^n\) (the set of all ordered \(n\)-tuples of real numbers) is a vector space over \(\mathbb{R}\)...
- C^n is a vector space.Theorem: \(\mathbb{C}^n\) (the set of all ordered \(n\)-tuples of complex numbers) is a vector space over...
- F^n is a vector space.Theorem: For any field \(\mathbb{F}\), the set \(\mathbb{F}^n\) (all ordered \(n\)-tuples of elements from...
- The set of all functions on a set is a vector space.Theorem: Let \(X\) be any set and \(\mathbb{F}\) a field. The set of all functions \(f: X \to \mathbb{F}\), denoted...
- The solutions to a homogeneous system of linear equations is a vector space.Theorem: The set of solutions to a homogeneous system of linear equations \(A\mathbf{x} = \mathbf{0}\) is a vector...
- The solutions to a homogeneous linear differential equation is a vector space.Theorem: The set of solutions to a homogeneous linear differential equation is a vector space. For example, the...
- The set of all polynomials is a vector space.Theorem: The set of all polynomials with coefficients in a field \(\mathbb{F}\), denoted \(P\) or \(\mathbb{F}[x]\),...
- The set of all polynomials of degree at most n is a vector space.Theorem: The set \(P n\) of all polynomials of degree at most \(n\) with coefficients in \(\mathbb{F}\) is a vector...
- The set of m by n matrices is a vector space.Theorem: The set \(M {m \times n}(\mathbb{F})\) of all \(m \times n\) matrices with entries in \(\mathbb{F}\) is a...
- The set of all sequences is a vector space.Theorem: The set of all infinite sequences \((a 1, a 2, a 3, \ldots)\) with entries in a field \(\mathbb{F}\) is a...
- The crazy vector space is a vector space.The "crazy" vector space is an example that shows the operations of addition and scalar multiplication need not look...
- The row space of a matrix is a vector spaceTheorem: The row space of a matrix \(A\) (the span of its row vectors) is a vector space --- specifically, a subspace...
- The column space of a matrix is a vector spaceTheorem: The column space of a matrix \(A\) (the span of its column vectors) is a vector space --- specifically, a...
- The null space of a matrix is a subspace of R^n (or C^n).Theorem: The null space of an \(m \times n\) matrix \(A\) is a subspace of \(\mathbb{F}^n\): \[\text{null}(A) =...
- The left null space of a matrix is a subspace of R^m (or C^m).For real $A\in\mathbb R^{m\times n}$, the left null space is $\ker A^T\subseteq\mathbb R^m$; for complex Hermitian...
- The set of linear transformations between two vector spaces is a vector space.Theorem: The set of all linear transformations from a vector space \(V\) to a vector space \(W\) (over the same field...
No children match.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.