Linear and multilinear algebra; matrix theory Inner product spaces Concept

Gram–Schmidt process

Builds on Orthonormal family Linear independence Linear span

Summary

Let $V$ be an inner product space over $\R$ or $\C$ , and let $v_1, \dots, v_k \in V$ be linearly independent. The Gram–Schmidt process defines vectors $u_1, \dots, u_k$ by $u_1 = v_1$ and, for $2 \le j \le k$ , $u_j = v_j - \sum_{i=1}^{j-1} \frac{\langle v_j, u_i \rangle}{\langle u_i, u_i \rangle} u_i.$ Then the vectors $e_j = u_j / \norm{u_j}$ form an orthonormal family with the same span as $v_1, \dots, v_j$ for each $j$ .

Remarks

Each step subtracts from $v_j$ its projections on the earlier vectors $u_i$ . For $v_1 = (1, 1)$ and $v_2 = (1, 0)$ in $\R^2$ , it gives $u_1 = (1, 1)$ and $u_2 = (1/2, -1/2)$ . In floating point arithmetic, the modified Gram–Schmidt process is more stable. In matrix form, the process gives the QR decomposition.

Reading path

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Vector space definition
Inner product definition
Linear combination definition
Linear subspace definition
Orthogonal vectors definition
Linear independence definition
Linear span definition
Orthonormal family definition
Gram–Schmidt process concept

Where this leads

16 pages build on this one.

Tag	`0VT6P0GNQR`
URI	`m2://0VT6P0GNQR`
Kind	concept
Writing	W1 stated
MSC	15A63, 65F25
Uses	Orthonormal family, Linear independence, Linear span
Same as	Wikidata Q475239 Wikipedia Gram–Schmidt process

Gram–Schmidt process

Summary

Remarks

Reading path

Where this leads

Further reading

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