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MATH 345: Linear Algebra and Optimization

Sebastien Roch with Jacob Denson and Kaitlyn Phillipson

Section 4.5 Applications and computation recap

This section checks whether you can read the computations already used in Unit 4.
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Subsection Review checkpoints

Activity 4.5.1. Checkpoint: reading residual orthogonality.

If A.T @ r is nearly zero, what geometric condition has been checked? Does this mean \(\mathbf{r}=\mathbf{0}\text{?}\)
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Tags. [U4-LO4, U4-LO7 | C+T | Core]
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Activity 4.5.2. Checkpoint: interpreting a near-zero \(A^T\mathbf{r}\) output.

A computation gives \(A^T\mathbf{r}=(0.000001,-0.000003)\text{.}\) What geometric condition is being checked? Does it mean the residual vector is zero?
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Tags. [U4-LO4, U4-LO7 | C+T | Core]
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Exam skill. Given a design matrix and residual output, explain projection onto the column space without solving a full regression problem by hand. Tags. [U4-LO4, U4-LO5, U4-LO7 | C+M+T | Core]
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Subsection Linked notebook

Run Lab U4: Regression as projection
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sebroc.github.io/MATH345-Course-Materials/labs/#lab-u4
. Focus on the design matrix, fitted coefficients, residual, and the computation A.T @ r.
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For a quick reference on least-squares commands, transposes, matrix products, and numerical checks such as "close to zero," see the programming appendix sections B.10, B.4, and B.9.
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