Build spaces from linear combinations and measure their dimension
A vector space is a set whose elements can be added and scaled without leaving the set. The central questions are generative: Which vectors can a set produce? Which vectors are redundant? What is the smallest coordinate system that still describes every element?
Vectors as algebraic and geometric objects
Use components without losing the directional interpretation
Addition
Add corresponding components. Geometrically, place vectors head-to-tail or use the parallelogram rule.
Scalar multiplication
Scale length by |α| and reverse direction when α is negative.
Norm
The Euclidean norm ‖v‖₂=√(v·v) measures length; distance is ‖u−v‖.
Span and subspaces
Describe every vector reachable from a generating set
The span of {v₁,…,vₖ} is the set of all linear combinations c₁v₁+···+cₖvₖ. It is automatically a subspace. To test whether another set W is a subspace, verify that it contains the zero vector and is closed under vector addition and scalar multiplication.
Two vectors in R²
If u and v are nonparallel and nonzero, their span is the entire plane. If they are parallel, every combination remains on one line. If both are zero, the span contains only the zero vector.
Not every line or plane is a subspace
A line or plane must pass through the origin to be closed under scalar multiplication and contain the zero vector. A shifted line is an affine set, not a vector subspace.
Independence, basis, and dimension
Remove redundancy while preserving the span
Linear independence
Vectors v₁,…,vₖ are independent when c₁v₁+···+cₖvₖ=0 forces every coefficient to be zero.
Basis
A basis is both independent and spanning. Every vector in the space then has one unique coordinate representation relative to that basis.
Dimension
All bases of a finite-dimensional vector space have the same number of vectors. That number is the dimension.
Interactive vector and span laboratory
Visualize combinations and test candidate bases
2D linear-combination visualizer
Adjust u, v, α, and β. The green vector is αu+βv.
Candidate basis test in R³
Each column is a candidate basis vector. Full rank means the columns form a basis for R³.
Analyze the matrix to test independence and span.
The four fundamental subspaces
Connect matrix structure to domain and codomain geometry
| Subspace | Definition | Dimension | Lives in |
|---|---|---|---|
| Column space C(A) | Span of columns; all outputs Ax. | rank(A) | Rᵐ |
| Null space N(A) | All x satisfying Ax=0. | n−rank(A) | Rⁿ |
| Row space C(Aᵀ) | Span of rows. | rank(A) | Rⁿ |
| Left null space N(Aᵀ) | All y satisfying Aᵀy=0. | m−rank(A) | Rᵐ |
Orthogonal pairings
In Rⁿ, the row space is orthogonal to N(A). In Rᵐ, the column space is orthogonal to N(Aᵀ). These decompositions support least squares and projections.
Review questions
Test span, subspaces, basis, and dimension