Linear Algebra Jonny Evans May 18, 2021 Contents 1 Week 1, Session 1: Matrices and transformations 5 1.1 Matrices . .5 1.1.1 Vectors in the plane . .5 1.1.2 2-by-2 matrices . .6 1.1.3 Mnemonic . .7 1.2 Matrices: examples . .7 1.2.1 Vectors in the plane . .7 1.2.2 Example 1 . .8 1.2.3 Example 2 . .8 1.2.4 Useful lemma . .8 1.2.5 Example 3 . .9 1.2.6 Example 4 . .9 1.2.7 Example 5 . 10 1.2.8 Example 6 . 11 1.2.9 Outlook . 11 1.3 Bigger matrices . 11 1.3.1 Bigger matrices . 11 1.3.2 More examples . 13 2 Week 1, Session 2: Matrix algebra 15 2.1 Matrix multiplication, 1 . 15 2.1.1 Composing transformations . 15 2.1.2 Mnemonic . 15 2.2 Matrix multiplication, 2 . 16 2.2.1 Examples . 16 2.3 Matrix multiplication, 3 . 17 2.3.1 Multiplying bigger matrices . 17 2.4 Index notation . 18 2.4.1 Index notation . 18 2.4.2 Associativity of matrix multiplication . 19 2.5 Other operations . 19 2.5.1 Matrix addition . 19 2.5.2 Special case: vector addition . 20 2.5.3 Rescaling . 20 1 2.5.4 Matrix exponentiation . 20 3 Week 2, Session 1: Dot products and rotations 22 3.1 Dot product . 22 3.2 Transposition . 23 3.3 Orthogonal matrices . 24 3.4 Rotations . 26 3.4.1 Example 1 . 26 3.4.2 Example 2 . 26 3.4.3 Example 3 . 27 4 Week 2, Session 2: Simultaneous equations 29 4.1 Simultaneous equations and row operations . 29 4.1.1 Simultaneous equations . 29 4.1.2 Solving these equations . 29 4.1.3 Row operations . 30 4.2 Echelon form . 31 4.3 Reduced echelon form . 32 4.3.1 Definition . 32 4.3.2 Reason for the definition . 33 4.3.3 Another example . 33 4.3.4 Strategy . 34 4.4 Echelon examples . 34 4.4.1 Example 1 . 34 4.4.2 Example 2 . 35 4.4.3 Example 3 . 36 5 Week 3, Session 1: Geometry of simultaneous equations and inverses 37 5.1 Geometric viewpoint on simultaneous equations, 1 . 37 5.1.1 2 variables . 37 5.1.2 Lines in the plane . 38 5.2 Geometric viewpoint on simultaneous equations, 2 . 39 5.2.1 3 variables . 39 5.2.2 Higher dimensions . 40 5.3 Subspaces . 41 5.3.1 Linear subspaces . 41 5.3.2 Affine subspaces . 41 5.3.3 More properties . 42 5.4 Inverses . 42 5.4.1 When can you “divide” by a matrix? . 42 5.4.2 Bigger matrices . 43 5.5 Inverses: examples . 44 5.5.1 Method for finding the inverse . 44 5.5.2 Examples . 44 6 Week 3, Session 2: Inverses and determinants 46 6.1 Elementary matrices, 1 . 46 6.1.1 Elementary matrices of type I . 46 6.1.2 Elementary matrices of type II . 47 2 6.2 Elementary matrices, 2 . 47 6.3 Determinants . 48 6.3.1 Definition . 48 6.3.2 Signs . 49 6.3.3 Index notation . 50 6.3.4 Examples . 50 6.4 Properties of determinants . 51 6.4.1 Type I row operations . 52 6.4.2 Type III row operations . 53 6.4.3 Type II row operations . 53 6.5 Determinants: examples . 53 7 Week 4, Session 1: More about determinants 55 7.1 Further properties of determinants . 55 7.1.1 Proof of the invertibility criterion . 55 7.1.2 Proof of multiplicativity of the determinant . 55 7.2 Cofactor expansion . 56 7.2.1 Signs . 57 7.2.2 Examples . 57 7.3 Geometry of determinants, 1 . 59 7.3.1 2-by-2 determinants and area . 59 7.3.2 Proof of theorem . 60 7.4 Geometry of determinants, 2 . 61 7.4.1 Higher dimensions . 61 7.4.2 Tetrahedra/simplices . 61 7.4.3 Determinant as scale factor for volumes . 63 8 Week 4, Session 2: Eigenvectors and eigenvalues 64 8.1 Eigenvectors and eigenvalues . 64 8.1.1 Eigenvectors and eigenvalues . 64 8.1.2 The eigenvector equation . 64 8.2 Finding eigenvalues . 65 8.2.1 Characteristic polynomial . 65 8.2.2 Examples . 66 8.2.3 Proof of theorem . 66 8.3 Eigenexamples . 67 8.4 Eigenspaces . 69 9 Week 5, Session 1: Eigenapplications 71 9.1 Differential equations . 71 9.1.1 Sketch of the idea . 71 9.1.2 An example in detail . 72 9.2 Ellipses . 73 9.2.1 General equation of an ellipse . 74 9.2.2 Normal form for ellipses . ..