Vector spaces

Definition

A vector space V over a field F consists of a set of elements (called vectors) along with two operations (vector addition and scalar multiplication) that satisfy the following axioms:

1. Closure under addition: For all u, v ∈ V, u + v ∈ V
2. Commutativity: For all u, v ∈ V, u + v = v + u
3. Associativity: For all u, v, w ∈ V, (u + v) + w = u + (v + w)
4. Additive identity: There exists an element 0 ∈ V such that v + 0 = v for all v ∈ V
5. Additive inverse: For each v ∈ V, there exists an element -v ∈ V such that v + (-v) = 0
6. Closure under scalar multiplication: For all v ∈ V and α ∈ F, αv ∈ V
7. Distributivity over vector addition: For all α ∈ F and u, v ∈ V, α(u + v) = αu + αv
8. Distributivity over scalar addition: For all α, β ∈ F and v ∈ V, (α + β)v = αv + βv
9. Scalar multiplication associativity: For all α, β ∈ F and v ∈ V, α(βv) = (αβ)v
10. Scalar multiplication identity: For all v ∈ V, 1v = v

Subspaces

A subset W of a vector space V is a subspace if and only if it satisfies:

1. The zero vector is in W
2. W is closed under vector addition: For all u, v ∈ W, u + v ∈ W
3. W is closed under scalar multiplication: For all v ∈ W and α ∈ F, αv ∈ W

Testing if a set is a subspace

For a set defined by equations, here’s how to test if it’s a subspace:

1. Check if it contains the zero vector (0,0,…,0)
2. Check if it’s closed under addition and scalar multiplication

For sets defined by other equations, rewrite them to see if they can be expressed as homogeneous linear equations.

Basis and Dimension

A set of vectors {v₁, v₂, …, vₙ} is a basis for a vector space V if:

1. The vectors are linearly independent
2. They span the vector space V (every vector in V can be written as a linear combination of these vectors)

The dimension of a vector space is the number of vectors in its basis.

Finding a Basis for a Subspace

To find a basis for a subspace defined by equations:

1. Convert the equations to a homogeneous system Ax = 0
2. Find the general solution to this system
3. Express the general solution in parametric form
4. The coefficient vectors of the parameters form a basis

Example: Testing if a Set is a Subspace

For S = {(x, y) ∈ ℝ² | x + y - 1 = 0}:

1. Check zero vector: (0,0) doesn’t satisfy x + y - 1 = 0, so S is not a subspace.

For S = {(x, y) ∈ ℝ² | (2x - y)(2x + y) = 0}: This means either 2x - y = 0 or 2x + y = 0.

1. Contains zero: (0,0) satisfies both equations, so it contains zero.
2. Closure under addition: If u = (u₁,u₂) and v = (v₁,v₂) are in S:
   • If both satisfy the same equation (e.g., 2x - y = 0), then u + v also satisfies it.
   • If they satisfy different equations, u + v might not satisfy either equation. So S is not closed under addition and is not a subspace.

For S = {(x, y) ∈ ℝ² | 4x² - 4xy + y² = 0}: This can be rewritten as (2x - y)² = 0, meaning 2x - y = 0.

1. Contains zero: (0,0) satisfies 2x - y = 0, so it contains zero.
2. Closure under addition and scalar multiplication: Since 2x - y = 0 is a linear equation, S is closed under both operations. So S is a subspace, and a basis is {(1,2)} (one vector because it’s a line through the origin).

Example: Finding Bases for Subspaces

For S = {(x, y, z) ∈ ℝ³ | x + y - 4 = 0, z = 0}:

1. Not a subspace because (0,0,0) doesn’t satisfy x + y - 4 = 0.

For S = {(x, y, z) ∈ ℝ³ | x + y - 4 = 0, z = 1}:

1. Not a subspace because (0,0,0) doesn’t satisfy z = 1.

For S = {(x, y, z, t) ∈ ℝ⁴ | x - 3y + z = 0}:

1. Contains zero: (0,0,0,0) satisfies x - 3y + z = 0.
2. Closure: This is a linear equation, so S is closed under addition and scalar multiplication. Therefore, S is a subspace.

To find a basis:

• Rewrite as x = 3y - z, so x is dependent on y, z, and t is free.
• General solution: (3y-z, y, z, t) = y(3,1,0,0) + z(-1,0,1,0) + t(0,0,0,1)
• A basis is {(3,1,0,0), (-1,0,1,0), (0,0,0,1)}

Exercises

1. Check if the following sets are subspaces and, if there exist, find two bases for each subspace:

   • $S = {(x, y) \in \mathbb{R}^2

     x + y - 1 = 0}$

   • $S = {(x, y) \in \mathbb{R}^2

     (2x - y)(2x + y) = 0}$

   • $S = {(x, y) \in \mathbb{R}^2

     4x^2 - 4xy + y^2 = 0}$