Check linear independence and find basis for a set of vectors
Enter 2 vectors in R2
v₁
v₂
Enter 3 vectors in R2
v₁
v₂
v₃
Enter 3 vectors in R3
v₁
v₂
v₃
Result
Step-by-Step Derivation
Vector Basis Formula
Basis = { v₁, v₂, ..., vₖ } linearly independent
Span(v₁,...,vₙ) = all a₁v₁ + ... + aₙvₙ
Form matrix A = [v₁ v₂ ... vₙ]
Basis = pivot columns of A
A basis for a vector space is a set of linearly independent vectors that span the space. To find a basis from a set of vectors, form a matrix with vectors as columns and identify pivot columns.
⚠If vectors are linearly dependent, remove redundant vectors to get a basis. The basis vectors come from the original set, not from RREF.
What is Vector Basis?
A basis for a vector space is a set of linearly independent vectors that span the entire space. Every vector in the space can be uniquely expressed as a linear combination of the basis vectors.
Linear Independence
No vector can be written as a linear combination of the others. Only trivial solution to c₁v₁+...+cₖvₖ=0.
Spanning Set
Every vector in the space can be expressed as a linear combination of these vectors.
Dimension
Number of basis vectors equals the dimension of the space.
Unique Representation
Each vector has exactly one representation as basis combination.
Linear AlgebraData AnalysisComputer GraphicsSignal ProcessingMachine Learning
Frequently Asked Questions
What is vector basis?▼
A basis for a vector space is a set of linearly independent vectors that span the space. Every vector can be uniquely expressed as their linear combination.
How to check linear independence?▼
1. Form matrix with vectors as columns. 2. Compute RREF. 3. If all columns are pivot columns, vectors are independent.
How to find basis from vectors?▼
1. Form matrix with vectors as columns. 2. Find pivot columns in RREF. 3. Corresponding original vectors form a basis.
What is span of vectors?▼
Span is the set of all possible linear combinations of the vectors. Dimension = rank of matrix formed by vectors.
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