This linear combination calculator is used in mathematics, particularly in linear algebra and vector spaces, to help compute the result of combining multiple vectors or scalars through addition and multiplication operations.
Consider a scenario where you have two vectors:
- v₁ = (2, 3, 1)
- v₂ = (4, -1, 5)
To find their linear combination with coefficients a = 2 and b = 3, you would calculate:
2v₁ + 3v₂ = 2(2, 3, 1) + 3(4, -1, 5) = (4, 6, 2) + (12, -3, 15) = (16, 3, 17)Linear Combination Calculator
| Vector 1 | Vector 2 | Coefficient 1 | Coefficient 2 | Result |
|---|---|---|---|---|
| (1, 2, 3) | (4, 5, 6) | 2 | 3 | (14, 19, 24) |
| (-1, 0, 2) | (3, -2, 1) | 0.5 | -1 | (-3.5, 2, 0) |
| (2, -1, 4) | (0, 3, -2) | -2 | 1.5 | (-4, 5.5, -11) |
| (5, 5, 5) | (-1, -1, -1) | 1 | 10 | (-5, -5, -5) |
| (0.5, 1.5, 2.5) | (1, 2, 3) | 4 | -0.5 | (1.5, 5, 8.5) |
| (3, 1, 2) | (2, 4, 6) | 1 | 2 | (7, 9, 14) |
| (0, 0, 0) | (1, 1, 1) | 3 | 1 | (3, 3, 3) |
| (1, 2, 1) | (2, 0, 3) | -1 | 2 | (0, 2, 5) |
| (1, 1, 1) | (1, 1, 1) | 0.5 | 0.5 | (1, 1, 1) |
| (4, 0, -2) | (1, 3, 5) | 2 | -1 | (6, -3, 8) |
| (2, 3, 4) | (5, -1, 2) | 0 | 4 | (20, -4, 8) |
| (1, -1, 0) | (0, 2, 3) | 3 | 1 | (3, 5, 3) |
| (6, 7, 8) | (1, 1, 1) | -2 | 1 | (4, 5, 6) |
| (3, 3, 3) | (2, 2, 2) | 1 | 1 | (5, 5, 5) |
| (1, 2, 3) | (-1, 0, 1) | 3 | 2 | (0, 6, 9) |
| (0, 1, 2) | (3, 4, 5) | 1 | -1 | (-2, -3, -3) |
Linear Combination Formula
For a set of vectors v₁, v₂, …, vₙ and corresponding scalar coefficients a₁, a₂, …, aₙ, the linear combination is expressed as:
a₁v₁ + a₂v₂ + ... + aₙvₙThis formula can be expanded to show individual components:
(a₁x₁ + a₂x₂ + ... + aₙxₙ, a₁y₁ + a₂y₂ + ... + aₙyₙ, a₁z₁ + a₂z₂ + ... + aₙzₙ)Let’s combine three vectors:
- v₁ = (1, 2, 3)
- v₂ = (4, 5, 6)
- v₃ = (7, 8, 9)
With coefficients a₁ = 2, a₂ = -1, and a₃ = 3, the linear combination is:
2v₁ - 1v₂ + 3v₃ = 2(1, 2, 3) - 1(4, 5, 6) + 3(7, 8, 9)
= (2, 4, 6) + (-4, -5, -6) + (21, 24, 27)
= (19, 23, 27)How to Find Linear Combination?
Identify the Vectors: Start with the vectors you want to combine. Let’s say v1 = (2, 3) and v2 = (1, 4).
Set Up the Equation: If you want to find a linear combination that equals b = (5, 10), set up the equation: c1 * v1 + c2 * v2 = b
Formulate the System of Equations: This leads to a system of equations based on the components:
c1 * 2 + c2 * 1 = 5 (1)
c1 * 3 + c2 * 4 = 10 (2)Solve the System: Use methods such as substitution or elimination to solve for c1 and c2.
For example, from equation (1): c2 = 5 – 2 * c1 Substituting c2 into equation (2): 3 * c1 + 4 * (5 – 2 * c1) = 10
3 * c1 + 20 – 8 * c1 = 10
-5 * c1 = -10
c1 = 2 Substituting back to find c2: c2 = 5 – 2 * (2) = 1 Thus, c1 = 2 and c2 = 1 are the coefficients for the linear combination.
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