Unit 2 Anatomy of a linear map
A matrix acts on inputs by \(\mathbf{x} \mapsto A\mathbf{x}\text{.}\) In applications, \(A\mathbf{x}\) might represent measurements from features, predictions from parameters, or scores from a model layer. This unit studies its anatomy through the equations \(A\mathbf{x}=\mathbf{b}\) (including the important special case where \(\mathbf{b}=\mathbf{0}\)): which outputs are reachable, which directions are forgotten, when inputs are non-unique, and how subspaces, independence, rank, nullity, and inverses organize the answers.
Big question. Given a linear map, how do systems of equations reveal what it produces, what it forgets, and when it can be reversed?
Learning outcomes. By the end of this unit, students should be able to:
Core outcomes.
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U2-LO1. Translate among geometric descriptions, systems of equations, augmented matrices, and matrix equations \(A\mathbf{x}=\mathbf{b}\text{.}\)
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U2-LO2. Use row reduction to solve linear systems, classify solution sets as empty, unique, or infinite, and describe solutions parametrically.
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U2-LO3. Compute and interpret \(\operatorname{im}(A)\text{,}\) \(\operatorname{null}(A)\text{,}\) rank, and nullity as reachable outputs, forgotten directions, transmitted dimensions, and non-unique inputs.
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U2-LO4. Use span, subspace, linear independence, basis, coordinates, and dimension to describe subspaces efficiently.
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U2-LO5. Determine whether a square matrix is invertible using row reduction, rank, null space, independence, inverses, and determinant criteria; compute inverses when appropriate.
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U2-LO6. Identify redundant columns or features and explain how null-space directions create different inputs or parameter vectors with the same output.
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U2-LO7. Compute determinants using cofactor expansion and determinant properties, and use determinants to test invertibility.
Toolbox skills. Set up augmented matrices, perform elementary row operations, find REF and RREF, solve homogeneous and nonhomogeneous systems, find basic solutions, compute bases for null spaces and column spaces, compute rank and nullity, test independence, compute inverses, and use determinant tests for invertibility.
Application and computation outcomes. Interpret Lab U2
[U2-LO1, U2-LO2, U2-LO3, U2-LO5, U2-LO6, U2-LO7 | C+R+T | Core] by reading computations about reachable outputs, row reduction, plane intersections, homogeneous systems, forgotten directions, rank, null vectors, redundant features, determinants, and inverse checks.
