The document discusses topics in mathematics that are fundamental to architecture, including geometry, matrices, eigenvalues and eigenvectors, algebraic structures, and affine transformations. It covers finding eigenvalues and eigenvectors of matrices, properties of eigenvalues and eigenvectors, and defines key concepts like vector spaces, affine spaces, and examples of affine transformations. The document appears to be notes for a university exam covering these essential mathematical foundations of architecture.

1. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FUNDAMENTALS OF MATHEMATICS I

2. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

3. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

4. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
21st DECEMBER EXAM
15% Today’s THEORY
15% Line’s THEORY
70% GEOMETRY
TRUSSES (internal forces in bars: Maxwell Cremona
CABLES (shape and internal forces in cables: Funicular Polygons
Eigen values + Eigen vectors
Algebraic structures
2D+ 3D line equations
Triangle centers

5. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
MATRICES: EIGEN VALUES + EIGEN VECTORS
ALGEBRAIC STRUCTURES
VECTOR SPACE
AFFINE SPACE
AFFINE TRANSFORMATIONS (or affinities). Examples of Affinities.
EUCLIDEAN VECTOR SPACE
BASIS OF A VECTOR SPACE
LINEAR MAP (or linear transformation)
CHANGE OF AXES OF REFERENCE

6. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
FIRST MINUTE QUESTION
Username: First part of your CEU email
For example:
if your email is a.garcia245@uspceu.es
Your username will be a.garcia245
Password: SAME AS USERNAME

7. FIRST MINUTE QUESTION
Which of these sentences is FALSE?
The vector director of line r that includes H and L is (0, -60, 70)
A
B
C
D
Line r that includes H and L belongs to the plane ZY
The vector director (0, -70, 60) is perpendicular to line r that contains H and L
The vector director (70, 0 , 40) is perpendicular to line t that contains W and H

8. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

9. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

10. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

11. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
PREVIOUS CONCEPTS
Order of a matrix = number of ROWS x number of COLUMNS
Square nxn matrix is a matrix in which number of ROWS = number of COLUMNS = n
Every square matrix A of order n, works as an OPERATOR that TRANSFORMS every vector v in another vector w:

12. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

13. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

14. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS
EIGENVECTORS are non-zero vectors that are transformed in proportional ones (same direction, equal or different size)
EIGENVALUES (or characteristic values) are those scalars l.
If you multiply A times he eigenvector x, you get back that same vector, multiplied by a scalar

15. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS
EIGENVECTORS are non-zero vectors that are transformed in proportional ones (same direction, equal or different size)
EIGENVALUES (or characteristic values) are those scalars l.
If you multiply A times he eigenvector x, you get back that same vector, multiplied by a scalar
A matrix can have multiple eigenvalues
BUT NO MORE than its number of n
(number of rows or columns)
Every EIGENVALUE is associated
with a specific EIGENVECTOR

16. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS
Is vector x an eigenvector of matrix A?

17. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS
Is vector x an eigenvector of matrix A?
l EIGENVECTOR

18. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
HOW TO FIND EIGENVALUES and EIGENVECTORS
THE DETERMINANT
OF THIS MATRIX MUST BE 0

19. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVALUES and EIGENVECTORS?

20. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVALUES?

21. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVECTORS?

22. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVECTORS?

23. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVECTORS?

24. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
¿EIGENVECTORS?

25. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

26. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

27. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

28. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS

29. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS. PROPERTIES
- The matrix trace (addition of the main diagonal) is the same as the addition of all the eigenvalues.
- The product of the eigenvalues is the same as the matrix determinant
- If there exists one lineal combination between lines or columns, at least one of the eigenvalues of the matrix is 0

30. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EIGENVALUES and EIGENVECTORS
* IN A COORDINATE SYSTEM DEFINED BY ITS EIGENVECTORS
* THE COMPONENTS OF THIS DIAGONAL MATRIX
ARE THE EIGENVALUES

31. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
ALGEBRAIC STRUCTURES
VECTOR SPACE
AFFINE SPACE
AFFINE TRANSFORMATIONS (or affinities). Examples of Affinities.
EUCLIDEAN VECTOR SPACE
BASIS OF A VECTOR SPACE
LINEAR MAP (or linear transformation)
CHANGE OF AXES OF REFERENCE

32. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

33. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
• A vector space is an algebraic structure that is applied not only to what we know as vectors.
• It can also be considered a vector space of matrices, or polynomials, or successions...

34. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

35. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

36. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
An affine space is what is left of a vector space after one has
forgotten which point is the origin (or, in the words of the
French mathematician Marcel Berger, "An affine space is nothing
more than a vector space whose origin we try to forget about,
by adding translations to the linear maps”.

37. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

38. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY
EXAMPLES OF AFFINITIES

39. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

40. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

41. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

42. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

43. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

44. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY

45. FUNDAMENTOS MATEMÁTICOS DE LA ARQUITECTURA I
FUNDAMENTALS OF MATHEMATIC IN ARCHITECTURE I GEOMETRY