Linear Combination Calculator
Calculate linear combinations of vectors with custom coefficients.
Select Vector Space Dimension
Number of Vectors
How Linear Combinations Work
A linear combination of vectors v₁, v₂, ..., vₙ with coefficients c₁, c₂, ..., cₙ is:
\[ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n \]
Properties of Linear Combinations:
- Closure: The result is always a vector in the same space
- Commutativity: Order of addition doesn't matter
- Associativity: Grouping doesn't matter
- Distributivity: c(v₁ + v₂) = cv₁ + cv₂
Applications:
- Vector spaces and subspaces
- Linear independence
- Span of vectors
- Basis vectors
- Coordinate systems