Linear Algebra: Vector Spaces and Subspaces
What Actually is a Vector Space?
In our everyday math, “vectors” are often depicted as arrows on a coordinate plane. But in advanced mathematics, a vector is anything that can be added and scaled while obeying a specific set of rules.
A vector space (V) is essentially “a set of vectors in a certain environment” over a field of scalars (F) (which can be real numbers, complex numbers, or imaginary numbers).
Inside this space, we must be able to perform two fundamental operations:
- Vector Addition (u + v)
- Scalar Multiplication (k .u)
If these operations keep the result within the same environment and satisfy standard behavior, they are defined by a set of 10 axioms (rules). Let’s review them!
10 Axioms:
- Closure under addition: For any u, v in V, u + v must also be in V
- Commutativity: u + v = v + u
- Associativity: u + (v + w) = (u + v) + w
- Additive Identity: There is a zero vector 0 such that u + 0 = u
- Additive Inverse: For every u, there is a -u such that u + (-u) = 0
- Closure under scalar multiplication: For any c in F and u in V, c.u must also be in V
- Identity of Scalar Multiplication: 1.u = u
- Compatibility (Associativity) of Scaling: ab.u = a(b.u)
- Distributivity of Scalars: (a + b).u = a .u + b.u
- Distributivity of Vector Addition: a.(u + v) = a.u + a.v
Diverse Examples of Vector Spaces
Examples of abstract mathematical spaces that are legally vector spaces:
Subspaces
What if we only want to look at a portion (a subset) of a vector space? Does that subset also qualify as a vector space?
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That is called a subspace.
A subset S of R^n is a valid subspace if and only if it passes the 3-Step Subspace Test:
- Contains the Zero Vector: 0 in S.
- Closed under Addition: If you add any two vectors in the subset, the result is still in the subset.
- Closed under Scalar Multiplication: If you scale any vector in the subset, the result is still in the subset.
Visualizing the Subspace Tests
To make this intuitive, look at the visual comparison based on the sketches from the notes:
Why is S = { [x_1, x_2]^T in{R}² , x_1 ≥ 0 } NOT a Subspace?
- Zero vector? Yes, (0,0) is in there.
- Closed under addition? Yes, if you add two right-pointing vectors, they will continue to point to the right.
- Closed under scalar multiplication? NO! If you take a vector like x = [2, 1]^T and multiply it by a negative scalar (like -1), you get -x = [-2, -1]^T. Because x_1 becomes negative (which is outside our defined region x_1 ≥ 0), this fails the test! Thus, this subset is not a subspace.
Why is the Span of a Vector a VALID Subspace?
“span” of any vector is the set of all its linear combinations (all its scale factors). If we take the span of the vector [1, 1]^T (the line y = x), we get a valid subspace:
- Zero vector? Yes, 0.[1,1]^T = [0,0]^T.
- Closed under addition? Yes, adding any two points on the line keeps the sum on the line.
- Closed under scalar multiplication? Yes, multiplying a point on the line by any scalar (positive or negative) just slides it somewhere else on the same infinite line.
Therefore, any Span forms a perfectly valid subspace of R^n.
