1. Vector Space
A vector space (also called a linear space) is a collection of objects called vectors that can be added together and multiplied (“scaled”) by numbers, called scalars. Scalars are usually real numbers (ℝ), but they can also be complex numbers (ℂ), rational numbers (ℚ), etc., depending on the context.
For a set to be called a vector space, it must satisfy some rules (called axioms) related to addition and scalar multiplication. These include:
- Closure under addition and scalar multiplication
- Associativity and commutativity of vector addition
- Existence of a zero vector
- Existence of additive inverses
- Distributive properties of scalar and vector addition
- Multiplicative identity: 1 × v = v
If all these conditions are satisfied, we call the set a vector space.
Common Examples of Vector Spaces:
ℝⁿ: The set of all n-dimensional real number vectors (e.g., [1, 2], [0, -5, 3])- The set of all polynomials of degree ≤ n
- The set of all continuous functions (e.g., sin(x), x², eˣ)
2. Subspace
A subspace is a smaller vector space that lives inside a bigger one. More precisely, a subspace is a subset of a vector space that is itself a vector space under the same operations.
To check whether a subset W of a vector space V is a subspace, you just need to verify three simple properties:
- Zero Vector: The zero vector of V must be in W.
- Closed under addition: If u and v are in W, then u + v must also be in W.
- Closed under scalar multiplication: If v is in W and c is any scalar, then c·v must also be in W.
If a subset satisfies all three properties, it is a subspace.
Example:
In ℝ³ (3D space), the set of all vectors of the form [x, 0, 0] is a subspace because:
- The zero vector
[0, 0, 0]is included (when x = 0). - Adding two such vectors (like [1, 0, 0] + [2, 0, 0] = [3, 0, 0]) stays in the set.
- Multiplying by a scalar (like 3 × [1, 0, 0] = [3, 0, 0]) also stays in the set.
3. Linear Combinations & Span
A linear combination is when you take some vectors and combine them using scalar multiplication and addition. For example, if you have vectors v₁ and v₂, then:
c₁·v₁ + c₂·v₂
is a linear combination, where c₁ and c₂ are scalars. You can have more than two vectors, and the idea is the same.
The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it’s all the points you can reach by scaling and adding those vectors.
Why is the span important?
The span tells you what part of the vector space you can cover using the given vectors.
Example:
Let’s say you have vectors v₁ = [1, 0] and v₂ = [0, 1] in ℝ². Any vector in the 2D plane can be written as:
c₁·[1, 0] + c₂·[0, 1] = [c₁, c₂]
So the span of v₁ and v₂ is the entire 2D plane ℝ².
But if you only had one vector, say [1, 0], then its span would just be the x-axis—because no matter how you scale it, you can’t go “up” or “down.”