Linear Algebra Done Right¶
Axler, S. (2015). Linear Algebra Done Right. Springer.
Cited by¶
14 citations across 14 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Basis
- Mathematics — bases of vector spaces, orthonormal bases, bases of free modules, and bases for topologies; generating sets in algebra as a related, weaker notion.
This sourceStandard treatment of bases as minimal spanning / maximal independent sets, the invariance of dimension, coordinate representations, and change of basis / diagonalization.
- Mathematics — bases of vector spaces, orthonormal bases, bases of free modules, and bases for topologies; generating sets in algebra as a related, weaker notion.
- Linear Combination
- In mathematics and physics it is vector-space spans, polynomial and Fourier bases, eigen-decompositions, Taylor series, and the superposition of solutions to linear differential equations.
This sourceDefines linear combinations, spans, and bases as the foundational construction of vector-space theory.
- In mathematics and physics it is vector-space spans, polynomial and Fourier bases, eigen-decompositions, Taylor series, and the superposition of solutions to linear differential equations.
- Linear Independence
- In mathematics and physics it is the independence of vectors, the functional independence of constraints, basis sets in quantum mechanics, and degrees of freedom in mechanics.
This sourceDefines linear independence, the dimension invariant, and bases over a vector space.
- In mathematics and physics it is the independence of vectors, the functional independence of constraints, basis sets in quantum mechanics, and degrees of freedom in mechanics.
- Transformation
- The signature separates two representations (state A and state B) and names the rule and mechanism that moves from one to the other while honoring constraints, in the spirit of Axler's (2015) operator-centric framing of linear maps as the primary objects of study rather than incidental structures defined on vectors.
This sourceOperator-centric framing of linear transformations as primary mathematical objects, emphasizing structural pattern of map plus preserved properties over coordinate-based computation.
Supported in partVerified against the publisher's abstract
The publisher's description of Axler's book names understanding the structure of linear operators as its central goal, backing the operator-centric framing clause but not the article's own 'signature' definition.
“The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces.”
- The signature separates two representations (state A and state B) and names the rule and mechanism that moves from one to the other while honoring constraints, in the spirit of Axler's (2015) operator-centric framing of linear maps as the primary objects of study rather than incidental structures defined on vectors.
- Vector Space
- A Basis is \(\{1, x, x^2\}\) — three linearly independent directions whose combinations cover the whole population — so the Dimension is 3 and every polynomial collapses to a coordinate tuple \((a_0, a_1, a_2)\).
This sourceVector spaces, bases, linear maps, rank and kernel as basis-independent invariants — the degree-bounded polynomial space and differentiation as a linear map.
- A Basis is \(\{1, x, x^2\}\) — three linearly independent directions whose combinations cover the whole population — so the Dimension is 3 and every polynomial collapses to a coordinate tuple \((a_0, a_1, a_2)\).
Domain-specific¶
- Corank
- Dimension (vector space)
- Kernel
- First, the kernel controls injectivity: \(T\) is injective if and only if \(\ker(T) = \{0\}\) (or \(\{e\}\) in the group case) — a trivial kernel is the exact characterization of when the map loses no information
This sourceStates the linear-map injectivity criterion, that T is injective exactly when its null space is {0}. Axler's rank-nullity identity, dim V = dim null T + dim range T, for a linear map between finite-dimensional spaces.
Supported in partVerified against the work's full text
“The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors.”
- First, the kernel controls injectivity: \(T\) is injective if and only if \(\ker(T) = \{0\}\) (or \(\{e\}\) in the group case) — a trivial kernel is the exact characterization of when the map loses no information
- Linear map
- Nilpotent Operator
- It applies to finite-dimensional transformations and to bounded operators on infinite-dimensional spaces under the same uniform-power test.
This sourceThe book explicitly assumes finite dimension for those theorems.
- It applies to finite-dimensional transformations and to bounded operators on infinite-dimensional spaces under the same uniform-power test.
- Outer product
- Rational dependence
Mechanisms¶
- Independent-Seed Basis Extension
- The Steinitz exchange lemma
This sourceProves that any independent list in a finite-dimensional vector space extends to a basis and that every basis has length equal to the space's dimension.
- The Steinitz exchange lemma
- Vector Linear-Combination Construction
- A subtler trap is linear dependence: when constituents are not independent, the same composite is reachable by many different weight sets
This sourceShows that coefficient representations are unique exactly for linearly independent spanning lists.
- A subtler trap is linear dependence: when constituents are not independent, the same composite is reachable by many different weight sets
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 2 of 14 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.
See how references were verified.
Links previously used in the corpus¶
Before the registry existed this work was also linked 2 other ways.
Registry ID ref:a6dcaf9bf234 · see in the full table