TL;DR (First Principles): A vector space is where linear algebra happens. If you can: * Add any two elements, * Multiply them by any scalar, * And still stay inside the same set, Then you're in a vector space.
Examples
- \(\mathbb{R}^n\): Ordinary n-dimensional real space.
- Set of all \(n \times m\) real matrices.
- Set of all polynomials with real coefficients.
- Set of all continuous functions on an interval.
- Set of 2D vectors lying in a plane through the origin in \(\mathbb{R}^3\).
Formal Definition
A vector space \(V\) over a field \(\mathbb{F}\) (usually \(\mathbb{R}\) or \(\mathbb{C}\)) is a set equipped with two operations:
- Vector addition: \(\mathbf{u} + \mathbf{v} \in V\)
- Scalar multiplication: \(c\mathbf{v} \in V\), where \(c \in \mathbb{F}\)
These operations must satisfy 8 axioms (rules), such as:
- Closure under addition and scalar multiplication
- Associativity, commutativity of addition
- Distributivity of scalar multiplication over scalars and vectors
- Existence of zero vector and additive inverses
- Multiplying by 1 leaves vector unchanged
Intuition
If you can:
- Add any two elements,
- Multiply them by any scalar,
- And still stay inside the same set,
Then you're in a vector space.
Examples
- \(\mathbb{R}^n\): Ordinary n-dimensional real space.
- Set of all \(n \times m\) real matrices.
- Set of all polynomials with real coefficients.
- Set of all continuous functions on an interval.
- Set of 2D vectors lying in a plane through the origin in \(\mathbb{R}^3\).
Non-Examples
- Vectors with a non-standard addition (e.g., max instead of +)
- Vectors that can’t be scaled (e.g., integers aren't closed under division)
Why It Matters
Vector spaces provide the stage for linear algebra. Once you’re in a vector space:
- You can apply linear maps (transformations).
- You can define bases, dimension, and coordinates.
- Concepts like span, linear independence, subspaces, and projections make sense.
In short, vector spaces are where linear algebra happens.