Vector Subspaces

A vector subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication. In other words, a subspace inherits the structure of the larger vector space.

Let V be a vector space over a field F (such as ℝ or ℂ), and let W be a subset of V. Then W is a subspace of V if:

1. Zero Vector: The zero vector of V is in W.
2. Closed under Addition: For any u, v ∈ W, the sum u + v ∈ W.
3. Closed under Scalar Multiplication: For any u ∈ W and any scalar c ∈ F, the product c·u ∈ W.

These conditions ensure that W satisfies all the axioms of a vector space.

Examples of Vector Subspaces

• Trivial Subspace:
  • {0}: The set containing only the zero vector is always a subspace of any vector space.
• Entire Vector Space:
  • V itself is a subspace of V.
• Lines and Planes through the Origin in ℝ³:
  • Any line through the origin in ℝ³ is a one-dimensional subspace.
  • Any plane through the origin in ℝ³ is a two-dimensional subspace.
• Solution Sets to Homogeneous Linear Equations:
  • Consider the equation Ax = 0, where A is a matrix. The set of all solutions x forms a subspace called the null space or kernel of A.
• Column Space and Row Space:
  • The set of all linear combinations of the columns of a matrix A is the column space of A, a subspace of ℝⁿ if A has n rows.
  • Similarly, the row space is the set of all linear combinations of the rows of A.
• Polynomials of Degree ≤ k:
  • The set P_k of all polynomials of degree at most k is a subspace of the vector space of all polynomials.

Properties of Vector Subspaces

Some of the common properties of vector subspaces are:

• Containment of the Zero Vector:
  • Every subspace must include the zero vector of the parent vector space.
• Closure Under Addition and Scalar Multiplication:
  • If you add any two vectors in the subspace, the result is still within the subspace.
  • If you multiply any vector in the subspace by a scalar, the result remains in the subspace.
• Intersection and Union:
  • The intersection of any collection of subspaces is also a subspace.
  • However, the union of two subspaces is generally not a subspace unless one is contained within the other.
• Dimension:
  • The dimension of a subspace is the number of vectors in a basis for that subspace. It cannot exceed the dimension of the parent vector space.

How to Determine if a Set is a Subspace

To verify whether a subset W of a vector space V is a subspace, follow these steps:

1. Check for the Zero Vector:
   • Ensure that the zero vector of V is in W.
2. Check Closure Under Addition:
   • Take any two vectors u and v in W. Verify that u + v is also in W.
3. Check Closure Under Scalar Multiplication:
   • Take any vector u in W and any scalar c. Verify that c·u is also in W.

Alternatively, you can use the Subspace Test, which states that a non-empty subset W of V is a subspace if for any u, v ∈ W and any scalars a, b, the linear combination a·u + b·v ∈ W.

Read More,

• Vector Algebra
• Linear Algebra
• Vector and Scaler
• Difference Between Scalar, Vector, Matrix and Tensor