Introduction to Linear Algebra¶
Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Decomposition
- Disassembly may be irreversible (a burned bridge cannot be fully reassembled); decomposition assumes reversibility and information preservation, a property Strang (2016) formalizes in linear algebra through eigendecomposition and SVD as exact, invertible factorizations.
This sourceDevelops eigendecomposition and the singular value decomposition as exact matrix factorizations.
- Disassembly may be irreversible (a burned bridge cannot be fully reassembled); decomposition assumes reversibility and information preservation, a property Strang (2016) formalizes in linear algebra through eigendecomposition and SVD as exact, invertible factorizations.
- Eigenvalue And Eigenvector
- The structural insight is not the algebra (Av = λv) but the decomposition move: a complex action on a space can be reorganized around its preserved directions, with the action collapsing into a list of independent scalar gains.
This sourceStandard treatment of diagonalization — reorganizing a transformation around its eigenvectors so iteration collapses to independent scalar gains.
- The structural insight is not the algebra (Av = λv) but the decomposition move: a complex action on a space can be reorganized around its preserved directions, with the action collapsing into a list of independent scalar gains.
- Projection
- In linear algebra and geometry the prototype is orthogonal projection onto a subspace — least-squares regression projects the response onto the column space, principal-component analysis projects onto the leading eigendirections, Fourier projection decomposes a signal onto frequency basis functions — with idempotence as the formal signature.
This sourceDevelops orthogonal projection onto a subspace, the projection (hat) matrix, and the idempotence identity P^2 = P.
- In linear algebra and geometry the prototype is orthogonal projection onto a subspace — least-squares regression projects the response onto the column space, principal-component analysis projects onto the leading eigendirections, Fourier projection decomposes a signal onto frequency basis functions — with idempotence as the formal signature.
- Span
- The reachable closure — their span — is the set of all \(a(1,0,0) + b(0,1,0) = (a, b, 0)\), which is exactly the \(xy\)-plane: a proper subspace, not the whole ambient space.
This sourceStandard reference defining the span of vectors as the set of all their linear combinations, basis as a minimal spanning set, rank as the dimension of the span, and change of basis between equal-span generating sets.
- The reachable closure — their span — is the set of all \(a(1,0,0) + b(0,1,0) = (a, b, 0)\), which is exactly the \(xy\)-plane: a proper subspace, not the whole ambient space.
- Transformation
- The structured mapping of an input to an output where the output is the input restructured according to a rule, preserving certain properties while altering others, as Strang (2016) develops in the canonical linear-algebra treatment of transformations as structured maps between vector spaces.
This sourceCanonical undergraduate text developing linear transformation as a structured map between vector spaces that preserves linearity while restructuring representation.
- The structured mapping of an input to an output where the output is the input restructured according to a rule, preserving certain properties while altering others, as Strang (2016) develops in the canonical linear-algebra treatment of transformations as structured maps between vector spaces.
- Vector Space
- Vector-space reasoning supports several characteristic moves. Linear combination as inference: if outputs are linear in inputs, the output of any combination is the combination of outputs — the superposition principle.
This sourceSuperposition, projection, subspace decomposition, basis change, and the least-squares geometry of vector spaces.
- Vector-space reasoning supports several characteristic moves. Linear combination as inference: if outputs are linear in inputs, the output of any combination is the combination of outputs — the superposition principle.
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