Linear Algebra Done Right

Axler, S. (2015). Linear Algebra Done Right. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2015

DOI

10.1007/978-3-319-11080-6

ISBN

9783319110790

Link

https://doi.org/10.1007/978-3-319-11080-6

Cited from

mathematics

Cited by

9 citations across 9 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.

• 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.

• 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.

• 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.

• Vector Space
  • A change of basis (say to a Legendre basis) changes the matrix entries — coordinate artifacts — but the rank of \(D\) (which is 2) and its kernel (the constants) survive: those are intrinsic.

    This sourceVector spaces, bases, linear maps, rank and kernel as basis-independent invariants — the degree-bounded polynomial space and differentiation as a linear map.

Domain-specific

• Corank
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Dimension (vector space)
  • … and a convenient observable transfers from R cubed over R has dimension three because the three coordinate unit vectors form a basis. to A proof distinguishes algebraic Hamel dimension from topological notions and specifies the scalar field, since the same set can have different dimension over different fields..
• Linear map
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Outer product
  • The coordinate outer product u vᵀ has entries (u vᵀ)ij=u_i v_j and rank at most one.

Verification

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Links previously used in the corpus

Before the registry existed this work was also linked 1 other way.

• https://link.springer.com/book/10.1007/978-3-319-11080-6 ×3

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