Calculator
Example data table
| v | Basis vectors | Projection p | ‖r‖ |
|---|---|---|---|
| [3, 1, 2] | [1,0,0], [0,1,0] | [3, 1, 0] | 2 |
| [2, -1, 4] | [1,1,0], [0,0,1] | [0.5, 0.5, 4] | 2.1213 |
| [1, 2] | [1,0] | [1, 0] | 2 |
Formula used
Let the subspace be spanned by basis vectors placed as columns in a matrix A (size n×k), and let v be the input vector.
- Orthonormal basis: p = A(Aᵀv).
- General basis: Solve (AᵀA + λI)c = Aᵀv, then p = Ac. The residual is r = v − p.
How to use this calculator
- Enter vector v with comma or space separators.
- Enter each basis vector on a new line.
- Keep all vectors the same dimension n.
- Use λ for stability if results look unstable.
- Click Compute to see projection, coefficients, and residuals.
- Download CSV or PDF to share calculations.
Why projection matters for accuracy
Subspace projection converts a vector into its closest approximation inside a chosen span. This calculator forms matrix A from your basis vectors, then computes the projection p and residual r = v − p. The norms ‖v‖, ‖p‖, and ‖r‖ summarize captured energy versus leftover error. A smaller ‖r‖ means the subspace explains more of v. The angle between v and p highlights alignment and interpretability. Use exports to document results for reviews later.
How the basis affects stability
Basis quality controls numerical stability. If columns of A are orthonormal, the shortcut p = A(Aᵀv) is fast and typically robust. With a general basis, the calculator solves (AᵀA + λI)c = Aᵀv and returns coefficients c. When AᵀA is nearly singular, tiny input noise can swing c dramatically. Regularization λ reduces sensitivity by damping unstable directions and improving invertibility. Start with λ = 1e-8, increase only if warnings appear. During computations.
Reading coefficients and residuals
Coefficients provide interpretable coordinates in your spanning set. Large coefficients can indicate redundancy, scaling issues, or an ill conditioned basis. Compare coefficient patterns while adjusting the basis; stable, moderate values are usually healthier. The residual vector r shows what the subspace cannot represent. Inspect r component by component to locate consistently missed dimensions. For iterative modeling, minimizing ‖r‖ across candidate subspaces supports objective feature selection and model justification. for consistent model comparison.
Where projections show up in practice
Projections appear in least squares fitting, dimensionality reduction, and orthogonal decomposition. In regression, Ac is the fitted response within the column space of A. In geometry, p is the perpendicular drop onto the span, while r measures the orthogonal error. In signal processing, projecting onto a learned subspace separates structure from noise. Exporting CSV or PDF preserves inputs, coefficients, and diagnostics, supporting audits, tutoring, and reproducible experiments. across teams and repeat sessions.
A reliable workflow for daily use
Practical workflow: start with a minimal, well scaled basis, then verify outputs. Keep all vectors the same dimension and avoid mixing units across components. If you are confident the basis is orthonormal, enable the shortcut for speed and reduced rounding error. Otherwise leave it off and begin with λ near 1e-8. Increase λ only when coefficients oscillate or rank warnings appear, then recheck ‖r‖ and angle. Use the example table to validate installation quickly.
FAQs
1) What if my basis vectors are not independent?
Dependent vectors make AᵀA close to singular, so coefficients become unstable. Remove redundant vectors or increase λ slightly to stabilize the solve and recheck the residual norm.
2) When should I enable the orthonormal shortcut?
Enable it only if your basis vectors are mutually orthogonal and each has unit length. Otherwise, keep it off and use the general method for correct results.
3) What does a large residual norm mean?
A large ‖r‖ means the subspace does not capture much of v. Consider adding meaningful basis vectors, changing the subspace, or reviewing whether v is scaled consistently.
4) Why can coefficients look huge even when the projection seems reasonable?
An ill-conditioned basis can require large opposing coefficients that cancel in Ac. Improve conditioning by using fewer, better-scaled vectors, or apply a small λ.
5) How do I interpret the angle between v and p?
Smaller angles mean v is well aligned with the subspace and most energy is captured. Larger angles indicate v has substantial components outside the subspace.
6) Does λ change the “true” projection?
Yes, λ produces a stabilized, ridge-style projection when the basis is poorly conditioned. Keep λ tiny when possible, and use it mainly to prevent numerical blowups.