Introduction to Linear Algebra¶
Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
Cited by¶
9 citations across 9 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.
Domain-specific¶
- Image (of a Function)
- In linear algebra, the image of a linear map \(T : V \to W\) is the column space of the representing matrix, its dimension is the rank of \(T\), and the rank-nullity theorem \(\dim(\mathrm{im}(T)) + \dim(\ker(T)) = \dim(V)\) pairs the image with the kernel in an exact accounting of the map's structure
This sourceThe column space of a matrix has dimension equal to its rank r and the nullspace has dimension n - r, the matrix form of rank-nullity.
Supported in partVerified against the source
- In linear algebra, the image of a linear map \(T : V \to W\) is the column space of the representing matrix, its dimension is the rank of \(T\), and the rank-nullity theorem \(\dim(\mathrm{im}(T)) + \dim(\ker(T)) = \dim(V)\) pairs the image with the kernel in an exact accounting of the map's structure
- Shear mapping
Mechanisms¶
- Span Membership Certificate
- Its strength is decisiveness for the positive case: a valid certificate ends the argument about whether a specific target is reachable, and it does so in a form a skeptic can check cheaply.
This sourceCoefficients that reproduce a target as a linear combination certify span membership and can be checked by direct substitution.
- Its strength is decisiveness for the positive case: a valid certificate ends the argument about whether a specific target is reachable, and it does so in a form a skeptic can check cheaply.
Verification¶
Does it exist? Not checked yet. This entry carries no identifier to resolve. It was extracted from the citation as written in the article, normalized, and deduplicated against the rest of the registry.
Does it back the claim? Read against the text for 1 of 9 citations: 1 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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Links previously used in the corpus¶
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