Gram Schmidt Calculator - Orthogonal and Orthonormal Basis

A Gram Schmidt calculator turns a list of 2 or 3 linearly independent vectors in 2D or 3D into an orthogonal set and then into an orthonormal basis for the same subspace. Use it to check the orthogonality of basis vectors in a class problem, to clean up a basis before a QR factorization, or to build a stable coordinate frame from measured directions.

• • Linear algebra homework: Walk through orthogonal and orthonormal basis steps for two or three input vectors in R^2 or R^3.
• • QR decomposition cross-check: Read the Q columns from e_i and the R diagonal from the Norms output to confirm a hand QR factorization.
• • Stable coordinate frames: Convert measured directions into a mutually perpendicular frame for graphics, robotics, or finite element work.
• • Linear dependence check: Submit a candidate basis; a linear-dependence error means the input does not span the claimed dimension.

Gram-Schmidt is a constructive proof that every finite-dimensional inner product space has an orthonormal basis. Given linearly independent vectors v_1, ..., v_n, the process produces u_1, ..., u_n that span the same subspace with every pair perpendicular, and then divides by length to give an orthonormal e_1, ..., e_n. The modified variant re-orthogonalizes each candidate vector against the most recent basis vector, which keeps the result stable for nearly parallel inputs.

When you also need the Euclidean length of each input or output vector alongside the basis, the Vector Magnitude Calculator covers the per-vector norm calculation that the Gram-Schmidt step implicitly reuses.

The calculator parses the textarea into input vectors, validates the count and dimension, runs modified Gram-Schmidt to produce the orthogonal u_i, and divides each u_i by its norm to produce the orthonormal e_i. The first orthogonal vector is the first input vector, and each later one is the next input vector minus its projection onto every earlier orthogonal vector.

• v_i: input vector at position i in the user-supplied list, 2D or 3D
• u_i: orthogonal vector at position i, output of the modified Gram-Schmidt recurrence
• e_i: orthonormal vector at position i, equal to u_i divided by its Euclidean norm
• r_ij: inner product (v_i . u_j) divided by (u_j . u_j); the diagonal r_ii equals ||u_i||, which appears in the Norms output

The recurrence comes from the projection formula proj_u(v) = ((v . u) / (u . u)) * u. Subtracting the projection leaves a vector perpendicular to u, so removing all earlier projections makes u_i perpendicular to the entire basis.

According to Wolfram MathWorld, the Gram-Schmidt process orthogonalizes a set of vectors by repeatedly subtracting the projection of the next vector onto the orthogonal basis built so far.

The projection coefficient (v_i . u_j) / (u_j . u_j) in the Gram-Schmidt recurrence is built from inner products, and the Dot Product Calculator computes the same dot product v . u for any pair of vectors you want to check by hand.

Four ideas carry the entire algorithm. Understanding them removes the guesswork from both the calculation and the interpretation of the result.

A quick way to remember the recurrence is to read it as 'start with v_i, then remove every earlier direction from it.' The output u_i has no component along any of the previous u_j.

Cross product is a different way to build a perpendicular vector in 3D, and the Cross Product Calculator is useful when you want to confirm that a Gram-Schmidt u_i matches the cross product of two others.

Paste the vectors, read off the orthogonal u_i, and then the orthonormal e_i. The same flow works for homework answers or QR extraction.

1. 1 List the input vectors: Write each vector on its own line, separating components with commas or spaces. Two vectors for a 2D or 3D pair, three for a full 2D or 3D basis.
2. 2 Match the dimension: Use 2 components per line for 2D, 3 for 3D. Mixed dimensions are rejected; for 2D vectors leave any third slot empty or set it to 0.
3. 3 Read the orthogonal set: The first output lists u_1, u_2, u_3, the orthogonal but unnormalized basis. Each u_i is perpendicular to every earlier u_j and the set spans the same subspace as the input.
4. 4 Read the orthonormal set: The second output lists e_1, e_2, e_3, the orthonormal basis. Every e_i has length 1, and the set is the Q matrix in a QR factorization.
5. 5 Use the norms row as R: The third output gives ||u_1||, ||u_2||, ||u_3||. These are the diagonal entries of the R matrix, and the off-diagonal R entries are the projection coefficients (v_i . u_j) / (u_j . u_j) computed during the recurrence.

An engineer has three measured directions (1, 1, 0), (1, 0, 1), (0, 1, 1) for a coordinate frame and wants to confirm it is orthogonal. Pasting the three vectors returns u_1, u_2, u_3 that are mutually perpendicular, and the e_i row gives the unit-length frame for downstream use.

Once you have the Q columns and the upper-triangular R row, the next step in many linear-algebra workflows is to put them to work on a 2x2 or 3x3 system, and the System of Equations Calculator runs Cramer's rule and Gaussian elimination on coefficients of the same dimension to reach an equivalent triangular form.

A purpose-built Gram Schmidt calculator saves you from re-doing the same projection steps by hand and keeps the bookkeeping visible at every iteration.

• • Visible intermediate steps: Both the orthogonal u_i and the orthonormal e_i are shown side by side, so you can read off the unnormalized basis for projection work and the unit basis for orthonormal frames from the same run.
• • Numerical stability: The modified recurrence reuses the just-updated vector for the next projection, which keeps roundoff in check when inputs are close to parallel.
• • QR decomposition shortcut: The Norms output is the diagonal of R, and the e_i row is the Q matrix. You can read off a full QR factorization without rewriting the algorithm.
• • Linear-dependence detection: The calculator checks every intermediate u_i for a near-zero norm. A linear-dependence error tells you the input vectors do not span the expected dimension.
• • Supports 2D and 3D inputs: Both 2D pairs and 3D triples use the same input format, so you can switch between class problems and real coordinate frames without re-formatting.

The same output doubles as a teaching aid and a workaday engineering tool: students see the projection arithmetic, practitioners see the QR-style R row and the Q columns at the same time.

With the orthonormal e_i basis in hand, point distances in the new frame collapse to a square root of summed squared components, and the 3D Distance Calculator runs that same Euclidean formula in the original coordinates whenever you want a direct sanity check against the projected distance.

A few decisions about inputs and tolerances change what the calculator returns. Knowing them helps you interpret borderline cases.

• • This calculator accepts 2 or 3 input vectors in 2D or 3D. For 4D or higher-dimensional inputs, the same recurrence still works but is not exposed by the page.
• • The result is rounded for display to four decimals. The internal arithmetic keeps full precision, but a hand calculation that rounds each intermediate value will not match the page output to more than a few digits.
• • The Norms output is the diagonal of R, but the off-diagonal projection coefficients are not displayed. Recover them from the recurrence if a full R is needed.

The recurrence and the output are equivalent to the QR factorization of the input matrix: e_i is the Q matrix and the Norms output is the diagonal of R, which is what makes Gram-Schmidt a workhorse algorithm in numerical linear algebra.

According to Wikipedia, dividing each orthogonal vector by its length gives an orthonormal set of vectors that span the same subspace as the original input vectors.

According to Khan Academy, the Gram-Schmidt process is the standard way to turn a list of linearly independent vectors into an orthonormal basis for the same subspace.

The QR factorization that this calculator implicitly computes is also the standard way to solve the least-squares problem in a linear regression fit, and the Linear Regression Calculator shows the same orthonormal-basis idea at work in that workflow.