Summary Mathematics II | VUB | 2025/26

2026




MATHEMATICS II
SUMMARY


KOEN HANEGREEFS
VUB

,Table of contents
Table of contents ..................................................................................................................................... 1
Chapter 1 - Linear Geometry ..................................................................................................................... 7
1.1 Vectors in ℝ𝒏.................................................................................................................................. 7
Intuition ............................................................................................................................................ 7
Definition ......................................................................................................................................... 8
Worked example ............................................................................................................................... 8
1.2 Lines, planes, hyperplanes .............................................................................................................. 8
Definition ......................................................................................................................................... 8
Rule ................................................................................................................................................. 8
Worked example ............................................................................................................................... 8
1.3 Dot product, length, angle ............................................................................................................... 8
Intuition ............................................................................................................................................ 8
Definition ......................................................................................................................................... 9
Rule ................................................................................................................................................. 9
Concept check ................................................................................................................................. 9
Chapter exercises ................................................................................................................................ 9
Chapter 2 - Vector Spaces ...................................................................................................................... 10
2.1 Vector space and subspace .......................................................................................................... 10
Intuition .......................................................................................................................................... 10
Definition ....................................................................................................................................... 10
Rule ............................................................................................................................................... 10
Worked example ............................................................................................................................. 10
2.2 Span, linear independence, basis .................................................................................................. 10
Intuition .......................................................................................................................................... 10
Definition ....................................................................................................................................... 11
Rule ............................................................................................................................................... 11
Worked example ............................................................................................................................. 11
2.3 Coordinate representation ............................................................................................................ 11
Definition ....................................................................................................................................... 11
Worked example ............................................................................................................................. 11


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, Chapter exercises .............................................................................................................................. 12
Chapter 3 - Maps Between Spaces.......................................................................................................... 13
3.1 Homomorphism and isomorphism ................................................................................................ 13
Intuition .......................................................................................................................................... 13
Definition ....................................................................................................................................... 14
Rule ............................................................................................................................................... 14
Worked example ............................................................................................................................. 14
3.2 Kernel, range, rank-nullity.............................................................................................................. 14
Definition ....................................................................................................................................... 14
Rule ............................................................................................................................................... 14
Worked example ............................................................................................................................. 14
3.3 Matrix representation .................................................................................................................... 15
Definition ....................................................................................................................................... 15
Worked example ............................................................................................................................. 15
Chapter exercises .............................................................................................................................. 15
Chapter 4 - Matrix Operations ................................................................................................................. 16
4.1 Addition, scalar multiple, product .................................................................................................. 16
Definition ....................................................................................................................................... 16
Rule ............................................................................................................................................... 16
Worked example ............................................................................................................................. 16
4.2 Inverse matrix ............................................................................................................................... 16
Definition ....................................................................................................................................... 16
Rule ............................................................................................................................................... 17
Worked example ............................................................................................................................. 17
Chapter exercises .............................................................................................................................. 17
Chapter 5 - Linear Systems ..................................................................................................................... 18
5.1 Gauss elimination ......................................................................................................................... 18
Intuition .......................................................................................................................................... 18
Definition ....................................................................................................................................... 18
Rule ............................................................................................................................................... 18
Worked example 5.A - Full Gauss workflow ...................................................................................... 18
5.2 Solution-set structure ................................................................................................................... 19


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, Rule ............................................................................................................................................... 19
Classification ................................................................................................................................. 19
Concept check ............................................................................................................................... 19
Chapter exercises .............................................................................................................................. 19
Chapter 6 - Determinants ....................................................................................................................... 20
6.1 Definition and small cases ............................................................................................................ 20
Definition ....................................................................................................................................... 20
Rule ............................................................................................................................................... 20
Worked example ............................................................................................................................. 20
6.2 Properties and use ........................................................................................................................ 20
Rule ............................................................................................................................................... 20
Worked example ............................................................................................................................. 21
Chapter exercises .............................................................................................................................. 21
Chapter 7 - Eigenvalues, Eigenvectors, Diagonalisation ........................................................................... 22
7.1 Eigenvalues and eigenvectors ........................................................................................................ 22
Intuition .......................................................................................................................................... 22
Definition ....................................................................................................................................... 23
Rule ............................................................................................................................................... 23
7.2 Diagonalisation ............................................................................................................................. 23
Definition ....................................................................................................................................... 23
Rule ............................................................................................................................................... 24
Worked example ............................................................................................................................. 24
Concept check ............................................................................................................................... 25
7.3 Change of basis and transition matrices ........................................................................................ 25
Intuition .......................................................................................................................................... 25
Definition ....................................................................................................................................... 25
Rule - Procedure 7-D: change of coordinates.................................................................................... 25
Rule - Procedure 7-E: change of basis for a linear map ...................................................................... 25
Worked example ............................................................................................................................. 25
Chapter exercises .............................................................................................................................. 25
Chapter 8 - Multivariable Calculus .......................................................................................................... 27
8.1 Partial derivatives and gradient ...................................................................................................... 27


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