Gram-Schmidt Calculator

The Gram-Schmidt algorithm takes a set of linearly independent vectors {x₁, x₂, ..., xₙ} and systematically generates an orthogonal set {v₁, v₂, ..., vₙ}. The process begins by setting the first orthogonal vector equal to the first original vector: v₁ = x₁. For each subsequent vector xₖ, we subtract its projections onto all previously computed orthogonal vectors.

The general formula for the k-th orthogonal vector is vₖ = xₖ - sum_{j=1}^{k-1} proj_{v_j}(xₖ), where the projection of x onto v is defined as proj_v(x) = ((x dot v) / (v dot v)) * v. To obtain an orthonormal basis, each resulting orthogonal vector vₖ is finally divided by its own Euclidean norm: uₖ = vₖ / ||vₖ||.

Let us find the second basis vector using the Gram-Schmidt method given two vectors: x₁ = (3, -2, 4) and x₂ = (4, 2, 1). First, we set our initial orthogonal vector v₁ equal to x₁, so v₁ = (3, -2, 4). Next, we compute the projection of x₂ onto v₁.

The dot product x₂ dot v₁ is (4)(3) + (2)(-2) + (1)(4) = 12 - 4 + 4 = 12. The dot product v₁ dot v₁ is (3)(3) + (-2)(-2) + (4)(4) = 9 + 4 + 16 = 29. Therefore, the projection is (12/29) * (3, -2, 4) = (36/29, -24/29, 48/29). Subtracting this projection from x₂ yields our second orthogonal vector: v₂ = (4, 2, 1) - (36/29, -24/29, 48/29) = (80/29, 82/29, -19/29).

Always verify that your initial vectors are linearly independent before starting the calculation, as dependent vectors will yield a zero vector during the process. Keep your fractions exact during manual checks, as rounding decimals early can compound errors in higher-dimensional spaces. Finally, double-check your work by computing the dot product between resulting orthogonal vectors; if the result is zero, your orthogonalization is correct.