Linear Algebra I
Vicky Neale
Michaelmas Term 2019
Contents
1 Introduction 2
2 Linear equations and matrices 3
2.1 Systems of linear equations . . . . . . . . . . . . . . . . . . . . 3
2.2 Solving systems of simultaneous linear equations using matrices 10
2.3 Elementary Row Operations (EROs) . . . . . . . . . . . . . . 13
2.4 Historical interlude 1 . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Reduced row echelon (RRE) form . . . . . . . . . . . . . . . . 16
2.6 Elementary matrices and the use of EROs to compute inverses 17
3 Vector spaces 21
3.1 Familiar vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 What is a vector space? . . . . . . . . . . . . . . . . . . . . . 21
3.3 Historical interlude 2 . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 More examples of vector spaces . . . . . . . . . . . . . . . . . 27
3.6 Subspaces of R, R2 , R3 . . . . . . . . . . . . . . . . . . . . . . 28
4 Bases 29
4.1 Spanning sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Linear independence . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 Row rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.6 Historical interlude 3 . . . . . . . . . . . . . . . . . . . . . . . 36
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Vicky Neale
Michaelmas Term 2019
Contents
1 Introduction 2
2 Linear equations and matrices 3
2.1 Systems of linear equations . . . . . . . . . . . . . . . . . . . . 3
2.2 Solving systems of simultaneous linear equations using matrices 10
2.3 Elementary Row Operations (EROs) . . . . . . . . . . . . . . 13
2.4 Historical interlude 1 . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Reduced row echelon (RRE) form . . . . . . . . . . . . . . . . 16
2.6 Elementary matrices and the use of EROs to compute inverses 17
3 Vector spaces 21
3.1 Familiar vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 What is a vector space? . . . . . . . . . . . . . . . . . . . . . 21
3.3 Historical interlude 2 . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 More examples of vector spaces . . . . . . . . . . . . . . . . . 27
3.6 Subspaces of R, R2 , R3 . . . . . . . . . . . . . . . . . . . . . . 28
4 Bases 29
4.1 Spanning sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Linear independence . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5 Row rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.6 Historical interlude 3 . . . . . . . . . . . . . . . . . . . . . . . 36
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