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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 13

13Duality and multilinear algebra

Beyond 21-242 This chapter goes beyond the course.

Dual spaces and dual maps, duality for inner product spaces, tensors, and the determinant revisited through alternating forms.

In this chapter

13.1Dual spaces
13.2Duality in inner product spaces
13.3Multilinear maps and tensors
13.4Alternating forms and the determinant revisited
13.5Proof patterns

Previous12.5Proof patterns Next13.1Dual spaces