Gram Schmidt Orthogonalization Solver

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Apply Gram–Schmidt to a list of vectors to produce an orthogonal (or orthonormal) set. Each new vector has all previous projection components removed: \( \mathbf{u}_k = \mathbf{v}_k - \sum_{j

Input vectors (one per line)

Visualization

Display decimals

Orthonormalize (normalize each \(\mathbf{u}_k\) to unit length) Show original vectors (dashed)

Formats accepted per line: [1 2 3], 1,2,3, (1; 2; 3). Optional labels like V1= are allowed. All vectors must have the same dimension. “Play” animates Gram–Schmidt once and stops automatically.

Ready

Before/after basis visualization

2D: drag to pan • wheel to zoom • double-click to reset. 3D: drag to rotate • Shift+drag to pan • wheel to zoom • double-click to reset. Play shows each subtraction of a projection and then stops automatically.

original \(\mathbf{v}_k\) (dashed) orthogonal \(\mathbf{u}_k\) orthonormal \(\mathbf{e}_k\) (if normalized)

Enter vectors and click “Calculate”.

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Theory

What Gram–Schmidt does

Gram–Schmidt orthogonalization is a standard procedure for turning a list of vectors \(\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_m\) in \(\mathbb{R}^n\) into a new list \(\mathbf{u}_1,\mathbf{u}_2,\dots,\mathbf{u}_m\) where the nonzero vectors are mutually orthogonal. “Orthogonal” means dot products are zero: \(\mathbf{u}_i\cdot\mathbf{u}_j=0\) when \(i\ne j\). If you also normalize each nonzero \(\mathbf{u}_k\) to unit length, you get an orthonormal list \(\mathbf{e}_k=\mathbf{u}_k/\lVert\mathbf{u}_k\rVert\) satisfying both orthogonality and \(\lVert\mathbf{e}_k\rVert=1\).

The key idea is “remove what is already explained by earlier directions.” The component of a vector \(\mathbf{v}\) in the direction of a nonzero vector \(\mathbf{u}\) is its projection: \[ \mathrm{proj}_{\mathbf{u}}(\mathbf{v})=\left(\frac{\mathbf{v}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\right)\mathbf{u}. \] The scalar coefficient \(\alpha=\frac{\mathbf{v}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\) tells you how much of \(\mathbf{u}\) is contained inside \(\mathbf{v}\). Subtracting this projection removes that “shadow” along \(\mathbf{u}\).

The algorithm step-by-step

Gram–Schmidt builds the orthogonal vectors sequentially:

Start with \(\mathbf{u}_1=\mathbf{v}_1\).
For \(k\ge 2\), remove from \(\mathbf{v}_k\) its projections onto the previously computed \(\mathbf{u}_1,\dots,\mathbf{u}_{k-1}\): \[ \mathbf{u}_k=\mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}(\mathbf{v}_k). \]
Optional normalization: if \(\mathbf{u}_k\neq \mathbf{0}\), set \(\mathbf{e}_k=\mathbf{u}_k/\lVert\mathbf{u}_k\rVert\).

When the input vectors are linearly independent, every \(\mathbf{u}_k\) is nonzero and the output is a full orthogonal (or orthonormal) set with the same span as the original vectors. If some input vector is dependent on earlier ones, its remainder becomes \(\mathbf{u}_k=\mathbf{0}\). That indicates no new direction was added.

Geometry and “shadow removal”

In 2D and 3D, you can visualize the projection \(\mathrm{proj}_{\mathbf{u}}(\mathbf{v})\) as the point where \(\mathbf{v}\) “casts a shadow” onto the line spanned by \(\mathbf{u}\). Subtracting the projection is like sliding \(\mathbf{v}\) perpendicular to that line until it has no component along \(\mathbf{u}\). Repeating this against each prior \(\mathbf{u}_j\) makes the new remainder orthogonal to all earlier directions.

Because the procedure is sequential, the order of the input vectors matters: the span of the final nonzero vectors is the same, but the intermediate orthogonal directions can change if you reorder the inputs.

University note: QR factorization connection

Gram–Schmidt is the classical route to the QR factorization. If you stack your input vectors as columns of a matrix \(A=[\mathbf{v}_1\ \mathbf{v}_2\ \cdots\ \mathbf{v}_m]\), then an orthonormal output can be collected as columns of a matrix \(Q=[\mathbf{e}_1\ \mathbf{e}_2\ \cdots]\). The coefficients used during the projection steps form an upper triangular matrix \(R\), and one obtains \(A=QR\). QR is central in numerical linear algebra, least squares problems, eigenvalue algorithms, and stability analysis. In practice, many implementations prefer a “modified Gram–Schmidt” or Householder reflections for better numerical stability, especially when vectors are nearly dependent.

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