Alphabeta MathMatrices, the Matrix of a Linear Map, and Change of Basis
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
Theorem
Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication
Corollary
Mₙ(F) is a ring under entrywise addition and matrix multiplication, including the zero ring M₀(F)
Definition
Matrix units Eᵢⱼ and the Kronecker delta
Lemma
EᵢⱼE_kℓ=δⱼₖE_iℓ
Corollary
Mₙ(F) is noncommutative for every n≥2
Definition
The transpose A^T of a matrix
Proposition
Transpose is linear and involutive, and (AB)^T=B^TA^T
Definition
The space L(V,W) of linear maps with pointwise addition and scalar multiplication
Proposition
L(V,W) is a vector space over the common scalar field
Definition
Invertible linear maps, linear isomorphisms, and inverse linear maps
Lemma
Identity maps and composites of linear maps are linear
Definition
Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
Theorem
[T(v)]_C=[T]_B^C[v]_B
Theorem
T↦[T]_B^C is a vector-space isomorphism L(V,W)≅ M_m× n(F)
Corollary
dim_F M_m× n(F)=mn and dim_FL(V,W)=(dim_FV)(dim_FW) for finite-dimensional V,W
Theorem
[S∘ T]_B^D=[S]_C^D[T]_B^C
Corollary
End_F(V) is a ring and matrix representation is a ring isomorphism End_F(V)≅ Mₙ(F)
Definition
Invertible matrices and the general linear group GLₙ(F)
Theorem
A square matrix is invertible exactly when its multiplication map is a linear isomorphism; matrices preserve inverses of linear isomorphisms
Corollary
Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension
Corollary
GLₙ(F) is a group under matrix multiplication, including the trivial group GL₀(F)
Definition
The change-of-basis matrix P_C leftarrowB=[id_V]_B^C
Theorem
[v]_C=P_C leftarrowB[v]_B and P_B leftarrowC=P_C leftarrowB⁻¹
Theorem
[T]_B'^C'=P_C'leftarrowC[T]_B^CP_B leftarrowB'
Definition
Similar matrices: B=P⁻¹AP for an invertible P
Theorem
Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar
Definition
The trace tr(A) as the sum of the diagonal entries
Proposition
Trace is a linear functional on Mₙ(F)
Theorem
For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
Corollary
Similar matrices have the same trace
Definition
The basis-independent trace of an endomorphism of a finite-dimensional vector space
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., §3C
- P. E. Leonard, Linear Algebra Notes, §3.4
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Exercises 14–15
- S. Schiavone, MIT 18.700 Day 9, §II.3.1
- S. Schiavone, MIT 18.700 Day 9, Proposition 26
- S. Schiavone, MIT 18.700 Day 9, Definition 21
- S. Axler, Linear Algebra Done Right, 4th ed., §3D
- S. Axler, Linear Algebra Done Right, 4th ed., Chapter 3
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Definition 3.31
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Matrix multiplication as composition
- S. Schiavone, MIT 18.700 Day 9, Proposition 29
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Dimension of matrix spaces
- S. Schiavone, MIT 18.700 Day 9, Corollary 27
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Definition 3.43 and Theorem 3.81
- S. Schiavone, MIT 18.700 Day 9, Theorem 35
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 3.80
- S. Schiavone, MIT 18.700 Day 9, Definition 33
- S. Axler, Linear Algebra Done Right, 4th ed., Theorem 3.86
- S. Schiavone, MIT 18.700 Day 9, invertibility section
- S. Schiavone, MIT 18.700 Day 9, Theorem 22
- S. Axler, Linear Algebra Done Right, 4th ed., §3D, result 3.82
- S. Schiavone, MIT 18.700 Day 9, Corollary 36
- S. Axler, Linear Algebra Done Right, 4th ed., §3D, results 3.81 and 3.84
- S. Schiavone, MIT 18.700 Day 9, change of basis
- S. Schiavone, MIT 18.700 Day 9, Definition 38
- S. Axler, Linear Algebra Done Right, 4th ed., §3D, result 3.84
- S. Schiavone, MIT 18.700 Day 9, similarity section
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.47
- S. Axler, Linear Algebra Done Right, 4th ed., §8D
- S. Axler, Linear Algebra Done Right, 4th ed., Theorem 8.49
- S. Axler, Linear Algebra Done Right, 4th ed., Theorem 8.50
- S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.51