Linear Algebra, 2020a

Topic: Fields ................................153
Topic: Crystals ..............................155
Topic: Voting Paradoxes .........................159
Topic: Dimensional Analysis .......................165
Chapter Three: Maps Between Spaces
I Isomorphisms .............................173
I.1 Definition and Examples ....................173
I.2 Dimension Characterizes Isomorphism .............183
II Homomorphisms ...........................191
II.1 Definition .............................191
II.2 Range Space and Null Space ..................199
III Computing Linear Maps .......................212
III.1 Representing Linear Maps with Matrices ...........212
III.2 Any Matrix Represents a Linear Map .............223
IV Matrix Operations ..........................232
IV.1 Sums and Scalar Products ....................232
IV.2 Matrix Multiplication ......................236
IV.3 Mechanics of Matrix Multiplication ..............244
IV.4 Inverses ..............................254
V Change of Basis ............................262
V.1 Changing Representations of Vectors ..............262
V.2 Changing Map Representations .................267
VI Projection ...............................275
VI.1 Orthogonal Projection Into a Line* ..............275
VI.2 Gram-Schmidt Orthogonalization* ...............280
VI.3 Projection Into a Subspace* ...................285
Topic: Line of Best Fit ..........................295
Topic: Geometry of Linear Maps ....................301
Topic: Magic Squares ...........................308
Topic: Markov Chains ..........................313
Topic: Orthonormal Matrices ......................319
Chapter Four: Determinants
I Definition ...............................326
I.1 Exploration* ...........................326
I.2 Properties of Determinants ...................331
I.3 The Permutation Expansion ..................337
I.4 Determinants Exist* .......................346
II Geometry of Determinants ......................355
II.1 Determinants as Size Functions .................355