Geometry, Eigenvalues, eigenvectors, linear transformations

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