2026
MATHEMATICS II
SUMMARY SOLUTIONS MANUAL
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
How to use this manual ............................................................................................................................ 6
Chapter 1 - Linear Geometry ..................................................................................................................... 7
E1.1 - vector arithmetic, length .......................................................................................................... 7
E1.2 - dot product, angle ................................................................................................................... 7
E1.3 - perpendicularity / parameter .................................................................................................... 8
E1.4 - line through points, membership .............................................................................................. 8
E1.5 - plane through three points ....................................................................................................... 8
E1.6 - hyperplane equation to parametrise ......................................................................................... 9
E1.7 - degenerate parametrisation ..................................................................................................... 9
E1.8 - hyperplane through points + line ............................................................................................ 10
Chapter 1 - Recap............................................................................................................................... 10
Chapter 2 - Vector Spaces ...................................................................................................................... 11
E2.1 - subspace test........................................................................................................................ 11
E2.2 - subspace test, fails................................................................................................................ 11
E2.3 - linear independence in R^3 .................................................................................................... 11
E2.4 - basis for solution space ......................................................................................................... 12
E2.5 - basis of P_2 ........................................................................................................................... 12
E2.6 - subspace check in P_3........................................................................................................... 13
E2.7 - spanning set vs basis ............................................................................................................. 13
E2.8 - coordinate representation ..................................................................................................... 14
Chapter 2 - Recap............................................................................................................................... 14
Chapter 3 - Maps Between Spaces.......................................................................................................... 15
E3.1 - linearity check ....................................................................................................................... 15
E3.2 - linearity fails .......................................................................................................................... 15
E3.3 - rank, nullity ........................................................................................................................... 15
E3.4 - kernel & range ....................................................................................................................... 16
E3.5 - matrix representation ............................................................................................................ 16
E3.6 - change of basis ..................................................................................................................... 17
E3.7 - homomorphism check ........................................................................................................... 17
1
, E3.8 - construct homomorphism ..................................................................................................... 18
Chapter 3 - Recap............................................................................................................................... 18
Chapter 4 - Matrix Operations ................................................................................................................. 19
E4.1 - matrix arithmetic ................................................................................................................... 19
E4.2 - matrix product ....................................................................................................................... 19
E4.3 - non-commutativity ................................................................................................................ 19
E4.4 - 2x2 inverse ............................................................................................................................ 20
E4.5 - 3x3 inverse via Gauss-Jordan ................................................................................................. 20
E4.6 - elementary matrices .............................................................................................................. 21
E4.7 - matrix-product dimensions .................................................................................................... 22
E4.8 - composition = product ........................................................................................................... 22
Chapter 4 - Recap............................................................................................................................... 23
Chapter 5 - Linear Systems ..................................................................................................................... 24
E5.1 - 2x2 Gauss ............................................................................................................................. 24
E5.2 - Gauss with free variable ......................................................................................................... 24
E5.3 - inconsistent system............................................................................................................... 24
E5.4 - particular + homogeneous ..................................................................................................... 25
E5.5 - Gauss-Jordan ........................................................................................................................ 25
E5.6 - kernel as solution set ............................................................................................................. 26
E5.7 - back-substitution mistake ..................................................................................................... 27
E5.8 - full Gauss workflow ............................................................................................................... 27
Chapter 5 - Recap............................................................................................................................... 28
Chapter 6 - Determinants ....................................................................................................................... 29
E6.1 - 2x2 determinant .................................................................................................................... 29
E6.2 - 3x3 Sarrus ............................................................................................................................. 29
E6.3 - Laplace expansion ................................................................................................................. 29
E6.4 - triangular determinant ........................................................................................................... 30
E6.5 - row reduction for determinant ................................................................................................ 30
E6.6 - inverse via adjugate ............................................................................................................... 31
E6.7 - cofactor sign ......................................................................................................................... 31
E6.8 - determinant + Cramer + interpretation.................................................................................... 31
Chapter 6 - Recap............................................................................................................................... 32
2
, Chapter 7 - Eigen-Theory ........................................................................................................................ 33
E7.1 - 2x2 eigenvalues ..................................................................................................................... 33
E7.2 - eigenvectors.......................................................................................................................... 33
E7.3 - characteristic polynomial ...................................................................................................... 33
E7.4 - diagonalisation ...................................................................................................................... 34
E7.5 - 3x3 eigenvalues ..................................................................................................................... 34
E7.6 - non-diagonalisable ................................................................................................................ 35
E7.7 - characteristic polynomial mistake .......................................................................................... 35
E7.8 - diagonalise + compute power ................................................................................................ 36
E7.9 - transition matrix & coordinate change .................................................................................... 36
E7.10 - diagonalisation via eigenbasis .............................................................................................. 37
Chapter 7 - Recap............................................................................................................................... 37
Chapter 8 - Multivariable Calculus .......................................................................................................... 38
E8.1 - partial derivatives .................................................................................................................. 38
E8.2 - gradient ................................................................................................................................ 38
E8.3 - homogeneity ......................................................................................................................... 38
E8.4 - critical points + Hessian ......................................................................................................... 39
E8.5 - directional derivative ............................................................................................................. 39
E8.6 - chain rule .............................................................................................................................. 40
E8.7 - Hessian sign rule ................................................................................................................... 40
E8.8 - contour-plot signs ................................................................................................................. 41
E8.9 - multivariable limit non-existence ........................................................................................... 41
E8.10 - chain rule in polar coordinates ............................................................................................. 41
Chapter 8 - Recap............................................................................................................................... 41
Chapter 9 - Optimisation ........................................................................................................................ 42
E9.1 - unconstrained min ................................................................................................................ 42
E9.2 - saddle point .......................................................................................................................... 42
E9.3 - Lagrange — single constraint ................................................................................................. 43
E9.4 - constrained - substitution & Lagrange .................................................................................... 43
E9.5 - open container cost min ........................................................................................................ 44
E9.6 - Hessian classification with leading minors ............................................................................. 44
E9.7 - Lagrange - missing condition .................................................................................................. 45
3
MATHEMATICS II
SUMMARY SOLUTIONS MANUAL
KOEN HANEGREEFS
VUB
,Table of contents
Table of contents ..................................................................................................................................... 1
How to use this manual ............................................................................................................................ 6
Chapter 1 - Linear Geometry ..................................................................................................................... 7
E1.1 - vector arithmetic, length .......................................................................................................... 7
E1.2 - dot product, angle ................................................................................................................... 7
E1.3 - perpendicularity / parameter .................................................................................................... 8
E1.4 - line through points, membership .............................................................................................. 8
E1.5 - plane through three points ....................................................................................................... 8
E1.6 - hyperplane equation to parametrise ......................................................................................... 9
E1.7 - degenerate parametrisation ..................................................................................................... 9
E1.8 - hyperplane through points + line ............................................................................................ 10
Chapter 1 - Recap............................................................................................................................... 10
Chapter 2 - Vector Spaces ...................................................................................................................... 11
E2.1 - subspace test........................................................................................................................ 11
E2.2 - subspace test, fails................................................................................................................ 11
E2.3 - linear independence in R^3 .................................................................................................... 11
E2.4 - basis for solution space ......................................................................................................... 12
E2.5 - basis of P_2 ........................................................................................................................... 12
E2.6 - subspace check in P_3........................................................................................................... 13
E2.7 - spanning set vs basis ............................................................................................................. 13
E2.8 - coordinate representation ..................................................................................................... 14
Chapter 2 - Recap............................................................................................................................... 14
Chapter 3 - Maps Between Spaces.......................................................................................................... 15
E3.1 - linearity check ....................................................................................................................... 15
E3.2 - linearity fails .......................................................................................................................... 15
E3.3 - rank, nullity ........................................................................................................................... 15
E3.4 - kernel & range ....................................................................................................................... 16
E3.5 - matrix representation ............................................................................................................ 16
E3.6 - change of basis ..................................................................................................................... 17
E3.7 - homomorphism check ........................................................................................................... 17
1
, E3.8 - construct homomorphism ..................................................................................................... 18
Chapter 3 - Recap............................................................................................................................... 18
Chapter 4 - Matrix Operations ................................................................................................................. 19
E4.1 - matrix arithmetic ................................................................................................................... 19
E4.2 - matrix product ....................................................................................................................... 19
E4.3 - non-commutativity ................................................................................................................ 19
E4.4 - 2x2 inverse ............................................................................................................................ 20
E4.5 - 3x3 inverse via Gauss-Jordan ................................................................................................. 20
E4.6 - elementary matrices .............................................................................................................. 21
E4.7 - matrix-product dimensions .................................................................................................... 22
E4.8 - composition = product ........................................................................................................... 22
Chapter 4 - Recap............................................................................................................................... 23
Chapter 5 - Linear Systems ..................................................................................................................... 24
E5.1 - 2x2 Gauss ............................................................................................................................. 24
E5.2 - Gauss with free variable ......................................................................................................... 24
E5.3 - inconsistent system............................................................................................................... 24
E5.4 - particular + homogeneous ..................................................................................................... 25
E5.5 - Gauss-Jordan ........................................................................................................................ 25
E5.6 - kernel as solution set ............................................................................................................. 26
E5.7 - back-substitution mistake ..................................................................................................... 27
E5.8 - full Gauss workflow ............................................................................................................... 27
Chapter 5 - Recap............................................................................................................................... 28
Chapter 6 - Determinants ....................................................................................................................... 29
E6.1 - 2x2 determinant .................................................................................................................... 29
E6.2 - 3x3 Sarrus ............................................................................................................................. 29
E6.3 - Laplace expansion ................................................................................................................. 29
E6.4 - triangular determinant ........................................................................................................... 30
E6.5 - row reduction for determinant ................................................................................................ 30
E6.6 - inverse via adjugate ............................................................................................................... 31
E6.7 - cofactor sign ......................................................................................................................... 31
E6.8 - determinant + Cramer + interpretation.................................................................................... 31
Chapter 6 - Recap............................................................................................................................... 32
2
, Chapter 7 - Eigen-Theory ........................................................................................................................ 33
E7.1 - 2x2 eigenvalues ..................................................................................................................... 33
E7.2 - eigenvectors.......................................................................................................................... 33
E7.3 - characteristic polynomial ...................................................................................................... 33
E7.4 - diagonalisation ...................................................................................................................... 34
E7.5 - 3x3 eigenvalues ..................................................................................................................... 34
E7.6 - non-diagonalisable ................................................................................................................ 35
E7.7 - characteristic polynomial mistake .......................................................................................... 35
E7.8 - diagonalise + compute power ................................................................................................ 36
E7.9 - transition matrix & coordinate change .................................................................................... 36
E7.10 - diagonalisation via eigenbasis .............................................................................................. 37
Chapter 7 - Recap............................................................................................................................... 37
Chapter 8 - Multivariable Calculus .......................................................................................................... 38
E8.1 - partial derivatives .................................................................................................................. 38
E8.2 - gradient ................................................................................................................................ 38
E8.3 - homogeneity ......................................................................................................................... 38
E8.4 - critical points + Hessian ......................................................................................................... 39
E8.5 - directional derivative ............................................................................................................. 39
E8.6 - chain rule .............................................................................................................................. 40
E8.7 - Hessian sign rule ................................................................................................................... 40
E8.8 - contour-plot signs ................................................................................................................. 41
E8.9 - multivariable limit non-existence ........................................................................................... 41
E8.10 - chain rule in polar coordinates ............................................................................................. 41
Chapter 8 - Recap............................................................................................................................... 41
Chapter 9 - Optimisation ........................................................................................................................ 42
E9.1 - unconstrained min ................................................................................................................ 42
E9.2 - saddle point .......................................................................................................................... 42
E9.3 - Lagrange — single constraint ................................................................................................. 43
E9.4 - constrained - substitution & Lagrange .................................................................................... 43
E9.5 - open container cost min ........................................................................................................ 44
E9.6 - Hessian classification with leading minors ............................................................................. 44
E9.7 - Lagrange - missing condition .................................................................................................. 45
3
