Abstract Vector Spaces
While coordinate vector spaces $F^n$ are the most concrete examples, the concept of a vector space extends to many other mathematical objects: functions, polynomials, sequences, matrices, and solutions to differential equations.
Key idea: If a set satisfies the vector space axioms (closure under addition and scalar multiplication, associativity, commutativity, identity elements, distributivity), then all theorems of linear algebra apply — regardless of what the "vectors" actually are.
This section covers:
- Formal definition and terminology
- Basic properties and consequences of the axioms
- Isomorphism — when two vector spaces are "the same" structurally
- Linear transformation algebra — operations on linear maps
- Eigenvalue theory for linear transformations (independent of matrix representation)
Parents
Children
- Linear combinationsDefinition: A linear combination of vectors \(\mathbf{v} 1, \mathbf{v} 2, \ldots, \mathbf{v} k\) in a vector space...
- SubspacesDefinition: A subset \(W\) of a vector space \(V\) is a subspace if \(W\) is itself a vector space under the same...
- Linear (in)dependenceDefinition: A set of vectors \(\{\mathbf{v} 1, \mathbf{v} 2, \ldots, \mathbf{v} k\}\) is linearly dependent if there...
- BasesDefinition: A basis of a vector space \(V\) is a set of vectors \(\{\mathbf{v} 1, \ldots, \mathbf{v} k\}\) that is:...
- Linear transformationsDefinition: A linear transformation (or linear map ) is a function \(T: V \to W\) between vector spaces over the same...
- Orthogonality and projectionOrthogonality means $\langle u,v\rangle=0$. For $u\ne0$, the projection of $v$ onto its line is $u\langle...
- Definition and terminologyDefinition: A vector space over a field $F$ is a set $V$ together with two operations: 1. Vector addition: $+: V...
- Basic propertiesTheorem (Basic Properties of Vector Spaces): Let $V$ be a vector space over $F$. Then: 1. $0v = 0$ — The scalar zero...
- IsomorphismDefinition: Two vector spaces $V$ and $W$ over the same field $F$ are isomorphic (written $V \cong W$) if there...
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