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- 1.1 definition: complex numbers
- 1.10 notation: n
- 1.11 definition: F^n, coordinate
- 1.13 definition: addition in F^n
- 1.14 commutativity of addition in F^n
- 1.15 notation: 0
- 1.17 definition: additive inverse in F^n, -x
- 1.18 definition: scalar multiplication in F^n
- 1.19 definition: addition, scalar multiplication
- 1.20 definition: vector space
- 1.21 definition: vector, point
- 1.22 definition: real vector space, complex vector space
- 1.24 notation: F^S
- 1.26 unique additive identity
- 1.27 unique additive inverse
- 1.28 notation: -v, w - v
- 1.29 notation: V
- 1.3 properties of complex arithmetic
- 1.30 the number 0 times a vector
- 1.31 a number times the vector 0
- 1.32 the number -1 times a vector
- 1.33 definition: subspace
- 1.34 conditions for a subspace
- 1.35 example: subspaces
- 1.36 definition: sum of subspaces
- 1.40 sum of subspaces is the smallest containing subspace
- 1.41 definition: direct sum
- 1.44 example: a sum that is not a direct sum
- 1.45 condition for a direct sum
- 1.46 direct sum of two subspaces
- 1.5 definition: additive inverse, subtraction, multiplicative inverse, division
- 1.6 notation: F
- 1.8 definition: list, length
- 1A R^n and C^n
- 1B Definition of Vector Space
- 1C Subspaces
- 2.1 notation: list of vectors
- 2.10 definition: polynomial, P(F)
- 2.11 definition: degree of a polynomial, deg p
- 2.12 notation: P_m(F)
- 2.13 definition: infinite-dimensional vector space
- 2.15 definition: linearly independent
- 2.17 definition: linearly dependent
- 2.19 linear dependence lemma
- 2.2 definition: linear combination
- 2.22 length of linearly independent list <= length of spanning list
- 2.25 finite-dimensional subspaces
- 2.26 definition: basis
- 2.28 criterion for basis
- 2.30 every spanning list contains a basis
- 2.31 basis of finite-dimensional vector space
- 2.32 every linearly independent list extends to a basis
- 2.33 every subspace of V is part of a direct sum equal to V
- 2.34 basis length does not depend on basis
- 2.35 definition: dimension, dim V
- 2.37 dimension of a subspace
- 2.38 linearly independent list of the right length is a basis
- 2.39 subspace of full dimension equals the whole space
- 2.4 definition: span
- 2.41 example: a basis of a subspace of P_3(R)
- 2.42 spanning list of the right length is a basis
- 2.43 dimension of a sum
- 2.6 span is the smallest containing subspace
- 2.7 definition: spans
- 2.8 example: a list that spans F^n
- 2.9 definition: finite-dimensional vector space
- 2A Span and Linear Independence
- 2B Bases
- 2C Dimension
- algebraic closure
- algebraic manipulation of exponents
- axiom of choice
- Chapter 1: Vector Spaces
- Chapter 2: Finite-Dimensional Vector Spaces
- Complexification of V
- data, stuff, structure, property
- Direct sum operates on vector spaces, not vectors
- equivalence relation
- exercise 2C.1
- exercise 2C.3
- exponentials
- Extemporanea
- fonts
- generalize the particular
- How can the empty set possibly satisfy additive inverse?
- How to define scaling for V^S?
- If b != 0 is false, doesn't that mean b = 0?
- Is R a subset of C? Is it a subspace?
- Isn't a relation just a boolean function?
- jds
- macros
- map notation
- meta: 1.35 example: subspaces
- non-constant polynomial with coefficients...
- proof by contradiction
- rationals
- relation
- RGB as vector space over GF(2^8)
- styles
- Subtleties when navigating levels of abstraction
- syntactic sugar
- Table of contents
- todo
- Visualizing exercise 2A.1
- What does it mean for a vector to contribute "new information" to a span?
- Which you should verify...