Notes for Linear Algebra Done Right

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jds 3rd January 2025 at 2:30am

Extemporanea

Chapter 1: Vector Spaces

1A R^n and C^n
1B Definition of Vector Space
1C Subspaces

Chapter 2: Finite-Dimensional Vector Spaces

2A Span and Linear Independence
2B Bases
2C Dimension

Chapter 3: Linear Maps

3A Vector Space of Linear Maps
3B Null Spaces and Ranges
3C Matrices
3D Invertibility and Isomorphisms
3E Products and Quotients of Vector Spaces
3F Duality

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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
3.1 definition: linear map
3.10 linear maps take 0 to 0
3.11 definition: null space, null T
3.13 the null space is a subspace
3.14 definition: injective
3.15 injectivity <-> null space equals 0
3.16 definition: range
3.18 the range is a subspace
3.19 definition: surjective
3.2 notation: L(V, W), L(V)
3.21 fundamental theorem of linear maps
3.22 linear map to a lower-dimensional space is not injective
3.24 linear map to a higher-dimensional space is not surjective
3.26 homogeneous system of linear equations
3.28 inhomogeneous system of linear equations
3.4 linear map lemma
3.5 definition: addition and scalar multiplication on L(V, W)
3.6 L(V, W) is a vector space
3.7 definition: product of linear maps
3.8 algebraic properties of products of linear maps
3A Vector Space of Linear Maps
3B Null Spaces and Ranges
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...

Table of contents