🔗 Linear Combination Calculator
Calculate linear combinations of vectors
How to Use This Calculator
Select Number of Vectors
Choose how many vectors you want to combine (2-5 vectors).
Choose Dimension
Select vector dimension (2D, 3D, 4D, or 5D).
Enter Vectors and Coefficients
Input each vector's components and corresponding coefficient.
Calculate
Get the linear combination result: c₁v₁ + c₂v₂ + ... + cₙvₙ
Formula
w = c₁v₁ + c₂v₂ + ... + cₙvₙ
Linear combination of vectors
Definition:
A linear combination is a vector obtained by multiplying each vector by a scalar (coefficient) and summing the results.
Example:
If v₁ = [a₁, a₂, a₃] and v₂ = [b₁, b₂, b₃], then c₁v₁ + c₂v₂ = [c₁a₁ + c₂b₁, c₁a₂ + c₂b₂, c₁a₃ + c₂b₃]
Properties:
- Span of vectors: all possible linear combinations
- Linear independence: no vector is a linear combination of others
- Basis: linearly independent vectors that span the space
About Linear Combination Calculator
The Linear Combination Calculator computes the result of combining multiple vectors with scalar coefficients. This fundamental operation in linear algebra is used to express new vectors as weighted sums of existing ones.
When to Use This Calculator
- Linear Algebra: Express vectors as combinations of basis vectors
- Span: Find if vectors span a space
- Linear Independence: Test if vectors are linearly independent
- Vector Spaces: Generate vectors from basis
- Transformations: Apply linear transformations
Why Use Our Calculator?
- ✅ Multiple Vectors: Combine up to 5 vectors
- ✅ Flexible Dimensions: 2D to 5D vectors
- ✅ Step-by-Step: Shows the calculation process
- ✅ Educational: Helps understand linear combinations
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Linear Combination: Weighted sum of vectors
- Span: Set of all linear combinations
- Basis: Minimal set of vectors that span a space
- Linear Independence: No vector can be expressed as combination of others
- Dimension: Number of vectors in a basis
Frequently Asked Questions
What is a linear combination?
A linear combination is a vector formed by multiplying each given vector by a scalar coefficient and adding the results: w = c₁v₁ + c₂v₂ + ... + cₙvₙ.
Can I combine vectors of different dimensions?
No, all vectors in a linear combination must have the same dimension. You can't add a 2D vector to a 3D vector.
What if all coefficients are zero?
If all coefficients are zero, the linear combination is the zero vector: 0v₁ + 0v₂ + ... = 0 (vector with all components zero).
How is this related to span?
The span of a set of vectors is the collection of all possible linear combinations. The calculator computes a specific combination, while span includes all such combinations.
What's the difference between linear combination and basis?
A linear combination is a single vector. A basis is a set of vectors from which all vectors in a space can be uniquely expressed as linear combinations.