Solve vector combinations accurately. Enter basis vectors, target values, inspect coefficients, verify span conditions, export reports, and visualize geometric relationships easily.
The graph displays basis vectors and the target vector from the origin.
| Dimension | v1 | v2 | v3 | Target | Expected Coefficients |
|---|---|---|---|---|---|
| 2D | (1, 0) | (0, 1) | Not used | (3, 4) | (3, 4) |
| 3D | (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | (2, -1, 5) | (2, -1, 5) |
| 3D | (1, 2, 0) | (0, 1, 1) | (2, 0, 1) | (5, 4, 3) | Computed by solver |
A target vector is written as a linear combination of basis vectors.
For 2D: T = c1v1 + c2v2
For 3D: T = c1v1 + c2v2 + c3v3
This creates a matrix equation Ac = b.
The calculator solves that system using Gaussian elimination.
If the determinant is zero, a unique linear combination does not exist.
When the determinant is nonzero, coefficients are unique.
Vector linear combinations are central to algebra, geometry, and applied mathematics. A vector can be described as a weighted sum of other vectors when it belongs to their span. Those weights are called coefficients. This calculator helps identify those coefficients clearly and checks whether the chosen vectors can generate the target vector.
In two dimensions, two independent vectors can form a basis for the plane. In three dimensions, three independent vectors can form a basis for space. When the selected vectors form a basis, every target vector in that space has one unique representation. That is why determinant checks matter. A zero determinant signals dependence, which removes uniqueness.
The calculator converts the vector equation into a matrix system. It then applies Gaussian elimination to solve for the unknown coefficients. This method is reliable, systematic, and widely used in academic and practical problem solving. It also reconstructs the target from the computed coefficients, so you can verify the final answer directly.
This tool is useful in coordinate transformations, span analysis, basis testing, linear algebra exercises, engineering models, graphics pipelines, and data methods. Students can test homework steps. Teachers can create examples. Analysts can verify whether a direction or state vector lies inside a generated space.
The included graph gives a visual sense of how the basis vectors and target vector relate. In simple cases, you can see how the target is reached by combining directional components. The export options help save worked results for reports, assignments, and revision notes. With one page, you get solving, checking, visualization, and documentation support.
A linear combination is a sum of vectors multiplied by scalar coefficients. It shows how one vector can be built from other vectors in the same space.
A nonzero determinant means the vectors are linearly independent. That guarantees a unique solution for the coefficients in a square system.
Yes. The calculator supports both 2D and 3D input. It automatically shows or hides the needed fields.
The calculator reports that a unique solution does not exist. Dependent vectors cannot create a proper basis for unique coefficients.
For a unique solution in standard coordinates, the number of basis vectors should equal the dimension. That produces a square system.
It shows the target rebuilt from the computed coefficients and basis vectors. Matching values confirm the solution is correct.
Yes. It is a standard algebra technique for solving matrix systems. It is accurate and widely taught.
You can save the solved coefficients, determinant, and vector summary. That helps with reports, notes, and record keeping.
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Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.