Skip to content

ThemeSystemLightDark

Text sizeSmallNormalLarge

WidthNormalWideFull

[ and ] go to the previous and next page.

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 4

4Dimension

Dimension, rank–nullity, rank inequalities, sums and direct sums, change of basis, quotients, and affine geometry.

In this chapter

4.1Dimension
4.2Rank–nullity
4.3Rank inequalities and the transpose
4.4Sums and direct sums
4.5Isomorphisms and change of basis
4.6Products and quotients Beyond 21-242
4.7Affine geometry Beyond 21-242
4.8Proof patterns

PreviousProblems for Chapter 3 Next4.1Dimension