Projection Calculator Purpose
A projection onto a span finds the closest vector inside a chosen subspace. Here the subspace is built from u1 and u2. The tool accepts any equal length vectors. It then builds the two column matrix A. The calculator solves the normal equations and returns the vector in the span that sits nearest to y.
Why This Matters
Projection is useful in statistics, engineering, graphics, and data fitting. It explains least squares in a compact way. The residual shows what part of y cannot be explained by the span. When the residual is perpendicular to both spanning vectors, the projection is correct. This check helps students avoid simple arithmetic mistakes.
What the Results Mean
The coefficients show how much of u1 and u2 are needed. The projection vector is c1 u1 plus c2 u2. The residual is y minus that projection. A small residual norm means y is already close to the span. A zero residual means y belongs to the span.
Handling Special Cases
Sometimes u1 and u2 are dependent. Then the two vectors form a line, not a plane. The calculator detects this with the Gram determinant. If the determinant is zero, it uses the nonzero spanning vector and computes a line projection. If both vectors are zero, no useful span exists.
Practical Tips
Use commas between entries. Keep vector lengths equal. Enter decimals when needed. Start with small examples before using long vectors. Compare the dot products against zero. Tiny nonzero values can happen because of rounding. The displayed values are rounded, but the calculations use full numeric values.
Accuracy Notes
The method relies on dot products and matrix inversion for a two by two Gram matrix. It is efficient and stable for normal classroom examples. For nearly dependent vectors, results may be sensitive. In that case, interpret the coefficients carefully and review the determinant.
Learning Value
This calculator also supports clear reporting. You can download results for notes, worksheets, or checking another method. The example table shows one complete input set. Use it to test the form first. Then replace the vectors with your own values and compare each output section. Save each output before changing inputs during practice sessions or assignments later.