Linear Algebra Technological Applications Edmar Candeia Gurjão
[email protected] Universidade Federal de Campina Grande
November 2017
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
1 / 12
Introdução
Edmar Candeia Gurjão
Department of Electrical Engineering - UFCG; Interests: Digital Signal Processing, Compressed Sensing;
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
2 / 12
Introdução
Edmar Candeia Gurjão
Analog domain: Calculus, Analog Computer. Digital domains: Linear Algebra. Digital Computer.
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
3 / 12
Introdução
Cursos de Álgebra Linear
• Subjects related to Linear Algebra MatriX Determinants Sistems of LInear Equations Vector Spaces Eigenvalues and eigenvectors Diagonalization SVD, LU, and other decompositions Principal Component Analysis (PCA) Independent Component Analysis (ICA) ....
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
4 / 12
Linear Algebra
Definition × Application
• Matrix Definition A = [aij ], aij ∈ R 1 ≤ i ≤ M, 1 ≤ j ≤ N
And? Ex.:
1 2 3 A= 4 5 6 7 8 9
Matrix multiplication: simple and easyl! https://docs.gimp.org/en/plug-in-convmatrix.html
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
5 / 12
Linear Algebra
Definition × Application
• Determinants Definition Given a swuare matrix AM×M its determinant is given by det(A) =
M X
aij δij ,
j=1
where ∆ij is the cofactor given by ∆ij = (−1)i+j det(A)ij , and det(A)ij is the determinant of the matrix obtained from A by removing row i and column j. And? Volume of a solid..
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
6 / 12
Linear Algebra
Definition × Application
• Inverse matrix Definition Given a matrix A, its inverse is a matrix A−1 such that A.A−1 = I, where I is the identity matrix. And? Criptography. Inverse propagation media.
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
7 / 12
Linear Algebra
Definition × Application
• Basis of a Vector Space Definition A set of vectors {v1 , v2 , ..., vn } is a basis for a space vector V if: A Linear Independent (LI) set; A set that geneartes the space V ; Obeys a set of axioms. And? Compact representation of a set of vectors.
Edmar Candeia Gurjão (UFCG/CEEI/DEE)
ATAL
November 2017
8 / 12