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1Vector spaces
2Linear maps and matrices
3Linear systems and row reduction
4Dimension
5Determinants
6Inner product spaces
7Eigenvalues and diagonalization
8The spectral theorems
9The singular value decomposition
10Generalized eigenspaces and Jordan form
11Markov chains and positive matrices
12Bilinear and quadratic forms
13Duality and multilinear algebra*
14Linear algebra and abstract algebra*
15Infinite-dimensional vector spaces*
16Applications to data and machine learning*
ASets, functions, and proofs
BComplex numbers
1. Overview
2. B.1Complex numbers
CThe Fundamental Theorem of Algebra*
1. Overview
2. C.1Proofs of the Fundamental Theorem of Algebra

Index of techniques
Practice
PDF

Chapter 2

2Linear maps and matrices

Linear maps, their matrices, matrix multiplication as composition, inverses, kernels and images, and the Fundamental Theorems.

In this chapter

2.1Linear maps
2.2Linear maps are determined by a basis
2.3Matrices of linear maps
2.4Matrix multiplication
2.5Invertibility
2.6Kernel and image
2.7The Fundamental Theorems
2.8Proof patterns

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