Gram-Schmidt Orthogonalization Calculator

Formula first

Overview

The Gram-Schmidt process is a systematic method for generating an orthogonal or orthonormal basis from a set of linearly independent vectors in an inner product space. It works by iteratively subtracting the projections of a vector onto the previously constructed orthogonal vectors to ensure the new vector is perpendicular to all predecessors.

Symbols

Variables

$u_k$ = Resulting Orthogonal Magnitude, $v_k$ = Input Vector Magnitude, $\sum$ $p_j$ = Sum of Projections

Apply it well

When To Use

When to use: Apply this algorithm when you need to construct an orthogonal basis for a subspace, which is essential for simplifying vector projections and performing QR decompositions. It assumes that the input set of vectors is linearly independent and that an inner product (like the dot product) is defined.

Why it matters: Orthogonal bases are computationally efficient because they eliminate cross-term interactions in matrix operations. This process is vital in computer graphics for coordinate transformations, in signal processing for noise reduction, and in numerical analysis to improve the stability of least-squares solutions.

Avoid these traps

Common Mistakes

• Using the original vectors instead of the newly found orthogonal vectors for subsequent projections.
• Calculation errors in the dot products used for scalar projections.

One free problem

Practice Problem

In a linear algebra exercise, a student is processing the second vector in a set. If the input vector vk has a component value of 12 and the sum of its projections onto the first orthogonal vector (projSum) is calculated as 4.5, find the corresponding component of the resulting orthogonal vector result.

Solve for: result

Hint: Subtract the sum of the projections from the original vector component.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Linear Algebra and Its Applications (5th ed.) by David C. Lay, Steven R. Lay, and Judi J. McDonald
2. Introduction to Linear Algebra (5th ed.) by Gilbert Strang
3. Wikipedia: Gram-Schmidt process
4. Linear Algebra and Its Applications by David C. Lay, 5th ed.
5. Introduction to Linear Algebra by Gilbert Strang, 5th ed.
6. Gram-Schmidt process (Wikipedia article title)
7. Linear Algebra and Its Applications by David C. Lay (5th Edition)
8. Numerical Linear Algebra by Lloyd N. Trefethen and David Bau III