Unit 6 Constraints and SVD
Constraints and matrix factorizations both ask for special directions. Lagrange multipliers find directions compatible with constraints. SVD finds the input directions where a matrix map stretches, weakens, or forgets information.
Big question. What directions matter most in a constrained problem or matrix map?
Learning outcomes. By the end of this unit, students should be able to:
Core outcomes.
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U6-LO1. Find absolute extrema on closed bounded regions by comparing interior and boundary candidates.
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U6-LO2. Set up and solve one-constraint Lagrange-multiplier problems, and interpret the condition \(\nabla f=\lambda\nabla g\) as parallel gradients.
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U6-LO3. Compute singular values and singular vectors in small examples and connect them to eigenvectors of \(A^TA\text{.}\)
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U6-LO4. Construct and interpret an SVD\begin{equation*} A=U\Sigma V^T, \end{equation*}including the roles of \(U\text{,}\) \(\Sigma\text{,}\) and \(V^T\text{.}\)
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U6-LO5. Use SVD to identify rank and orthonormal bases for the four fundamental subspaces.
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U6-LO6. Interpret singular values as stretch factors and as transmitted, weak, or forgotten directions.
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U6-LO7. Use truncated SVD or rank-\(k\) reconstruction to explain compression and dominant directions.
Toolbox skills. Compare interior and boundary candidates, set up Lagrange equations, solve small constrained optimization problems, compute \(A^TA\text{,}\) find singular values and singular vectors, read rank from SVD, identify fundamental subspaces, compute energy retained, interpret rank-\(k\) reconstruction, and read low-rank update shapes.
Application and computation outcomes. Interpret Unit 6 computation snippets and Lab U6
[U6-LO4, U6-LO5, U6-LO6, U6-LO7 | C+M+T | Core] by reading SVD shapes, singular values, singular-vector identities, rank and null directions, redundant features, energy retained, rank-\(k\) reconstructions, compression on a small matrix, and low-rank update shapes.
