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Vector Spaces

A vector space has dimension = size of a basis. Span = all linear combinations. Vectors are linearly independent iff no non-trivial combination gives zero. rank = dimension of column space.

Concept Fundamentals
# basis vectors
Dimension
all lin. combinations
Span
dim(column space)
Rank
c₁v₁+…+cₙvₙ=0⇒cᵢ=0
Independence
Analyze Vector SpaceDimension, basis, span, independence

Why This Mathematical Concept Matters

Why: Vector spaces underpin linear algebra: solving Ax=b, eigenvalues, and change of basis. Dimension = degrees of freedom.

How: Stack vectors as rows; row-reduce to RREF. Pivot columns = basis. Rank = # pivots. Null space from free variables.

  • ●rank + nullity = n (columns).
  • ●Basis = maximal linearly independent set.
  • ●Span = smallest subspace containing vectors.
Sources:Gilbert Strang, Linear AlgebraKhan Academy

Quick Examples — Click to Load

Vectors (rows)

v1=
v2=
v3=
vector_space.sh
CALCULATED
$ vector-space --vectors=3 --dim=3
Dimension
3
# Basis Vectors
3
Independent?
Yes
Rank
3
Dimension of Span

3

Linearly Independent?

Yes

Basis Vectors
b1 = (1.0000, 0.0000, 0.0000)
b2 = (0.0000, 1.0000, 0.0000)
b3 = (0.0000, 0.0000, 1.0000)
Share:
Vector Space
Dimension: 3 | Rank: 3
Independent: Yes
numbervibe.com/calculators/mathematics/linear-algebra/vector-space-calculator

Metrics (Bar)

Independence (Doughnut)

Calculation Steps

SETUP
Input vectors3 vectors in R^3
FORMULA
Form matrix (vectors as rows)Row reduce
RESULT
Rank3
RESULT
Dimension of span3
RESULT
# Basis vectors3
RESULT
Linearly independent?Yes

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

dim(span) = rank

⚡

Basis = min spanning set

Key Takeaways

  • • Dimension = rank = # of linearly independent vectors in a basis.
  • • Span = all linear combinations of the vectors.
  • • Basis = maximal linearly independent subset.
  • • Linear independence: c₁v₁ + ... + cₙvₙ = 0 ⟹ all cᵢ = 0.
  • • Rank = dimension of row space = dimension of column space.

Did You Know?

⚡Row reduction (Gaussian elimination) finds rank in O(n³).Source: Complexity
🧠PCA finds orthogonal basis of maximum variance directions.Source: ML
📊Span of columns = column space; span of rows = row space.Source: Linear Algebra
🎮Graphics: 3 basis vectors define a coordinate frame.Source: Graphics
⏱️n vectors in Rⁿ are independent ⟺ matrix has full rank.Source: Properties
⚛️Quantum states live in Hilbert spaces (infinite-dimensional).Source: Physics

How It Works

1. Form matrix: Vectors as rows (or columns).

2. Row reduce: Gaussian elimination to row echelon form.

3. Rank: Number of pivot columns = dimension of span.

4. Basis: Original vectors corresponding to pivot columns.

dim(span) = rank = # pivot columns

Independent ⟺ rank = # vectors

Expert Tips

m vectors in Rⁿ

If m > n, vectors must be dependent.

Standard Basis

e₁,...,eₙ in Rⁿ always independent.

Zero Vector

Any set containing 0 is dependent.

Orthogonal

Non-zero orthogonal vectors are independent.

Comparison Table

FeatureThis CalculatorNumPyManual
Dimension & Rank✅✅⚠️
Basis identification✅❌⚠️
Bar & Doughnut charts✅❌❌
8 preset examples✅❌❌

FAQ

What is the dimension of a vector space?

Number of vectors in any basis. Equals the rank.

What is a basis?

Maximal linearly independent set that spans the space.

What is the span?

Set of all linear combinations c₁v₁ + ... + cₙvₙ.

When are vectors linearly independent?

When c₁v₁ + ... + cₙvₙ = 0 ⟹ all cᵢ = 0.

What is the rank of a matrix?

Dimension of row space = dimension of column space.

Can 4 vectors in R³ be independent?

No. Max 3 independent vectors in R³.

What is the null space?

Vectors v with Av = 0. Dimension = n - rank.

How does row reduction work?

Gaussian elimination: pivot, eliminate, repeat. Pivots = rank.

Stats

dim=rank
Dimension
O(n³)
Row reduction
m≤n
Max indep in Rⁿ
span⊆Rⁿ
Subspace

Sources

Disclaimer: For educational purposes. Uses JavaScript floating-point. Verify critical calculations independently.

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