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Intermediate Topic 30 of 51 Math for Computing

Orthogonality and Projections

Perpendicular vectors and closest-point maps.

How it works

Two vectors are orthogonal when their dot product is zero, meaning they are perpendicular. A set of vectors is orthonormal if they are orthogonal unit vectors, and orthogonal matrices preserve lengths and angles. The projection of a vector b onto a vector a is (a dot b over a dot a) times a, and projection onto a subspace gives the closest point in it, with the residual orthogonal to the subspace. The Gram-Schmidt process builds orthonormal bases, and the QR decomposition expresses a matrix with an orthogonal factor.

Where it is used

Least-squares regression as projection of data onto the column space, signal decomposition, graphics shadows and reflections, Fourier analysis and numerically stable algorithms rely on orthogonality.

Common pitfall

Gram-Schmidt in floating point loses orthogonality, so use modified Gram-Schmidt or Householder reflections. Orthogonal in function spaces needs an appropriate inner product. Mistaking uncorrelated for independent is a related error. Check projections by verifying that the residual is orthogonal to the subspace.

Where it fits in the course

Orthogonality and Projections is topic 30 of 51 in Math for Computing, rated Intermediate. It follows Singular Value Decomposition. The next topic is Derivatives.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Math for Computing quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Orthogonality and Projections?
Perpendicular vectors and closest-point maps. It belongs to Math for Computing and is an intermediate-level topic.
Where do people use Orthogonality and Projections?
Least-squares regression as projection of data onto the column space, signal decomposition, graphics shadows and reflections, Fourier analysis and numerically stable algorithms rely on orthogonality.
What is the most common mistake with Orthogonality and Projections?
Gram-Schmidt in floating point loses orthogonality, so use modified Gram-Schmidt or Householder reflections.
What should I learn before this topic?
Start with Singular Value Decomposition, the topic just before it in Math for Computing.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Math for Computing quiz.
Is Orthogonality and Projections hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Orthogonality and Projections?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Orthogonality and Projections?
Singular Value Decomposition, Derivatives, Eigenvalues and Eigenvectors are closely connected. The full list is in the related topics section of this page.
What should I study after Orthogonality and Projections?
Continue with Derivatives, the next topic in Math for Computing.

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