Kulum alin-11 jan2014

Kuliah Umum

Aljabar Linear

Rolly Rochmad [email protected]

Universitas Serang Raya11 Januari 2014

Aljabar Linear

System of Linear Equation & Matrices Determinant Vektor Ruang Dimensi 2 dan 3 Ruang Vektor Euclidean General Vector Space Inner Product Spaces Eigen Value dan Eigen Vector Linear Transformation Applications

Computer Graphics ( Visualization of a Three-Dimensional Object)

Pi Pj denotes that point Pi is connected to point Pj

Computer Graphics ( Visualization of a Three-Dimensional Object)It should be noticed that only the x and y-coordinates of the vertices are needed by the video display system to draw the view, because only the projection of the objectonto the xy-plane is displayed. However, we must keep track of the z-coordinates to carry out certain transformations discussed later.

Computer Graphics ( Visualization of a Three-Dimensional Object)We now show how to form new views of the object by scaling, translating, or rotating the initial view. We first construct a 3 x n matrix P, referred to as the coordinate matrix of the view, whose columns are the coordinates of the n points of a view:

Computer Graphics ( Visualization of a Three-Dimensional Object)

Scaling If a point Pi has coordinates (Xi, Yi, Zi) in the original view, it is to move to a new point Pi’ with coordinates (Xi, Yi, Zi) in the new view.

Pi’ = SPi

ex : scaled by =1, =0.5, =3

Computer Graphics ( Visualization of a Three-Dimensional Object)Translation to change an existing view so that each point Pi with coordinates (Xi, Yi, Zi) moves to a new point Pi‘ with coordinates (Xi+ Xo, Yi+ Yo, Zi+ Zo)

ex : scaled by Xo =1.2 , Yo =0.4, Zo =1.7

Computer Graphics ( Visualization of a Three-Dimensional Object)Rotation P’ = RProtated 90° about the x-axis

Computer Graphics ( Visualization of a Three-Dimensional Object)Rotation P’ = RProtated 90° about the y-axis

Computer Graphics ( Visualization of a Three-Dimensional Object)Rotation P’ = RProtated 90° about the z-axis

Computer Graphics ( Visualization of a Three-Dimensional Object)Rotation rotate first about the x-axis through 30°, then about the y-axis through −70°, and finally about the z-axis through −27° P’ = RP =R1R2R3P

CRYPTOGRAPHY: Hill Ciphers

Z=0

Example

A system of cryptography in which the plaintext is divided into sets of n letters, each of which is replaced by a set of n cipher letters, is called a polygraphic system.

Hill ciphers is a class of polygraphic systems based on matrix transformations.

Ciphertext vector = chipper x plaintext vector = Ap

CRYPTOGRAPHY: Hill Ciphers

Z=0

Coding The Matrix: Linear Algebra Through Computer Science Applications

• Prof. Philip N Klein of the Brown University• http://codingthematrix.com/• http://cs.brown.edu/courses/cs053/current/lectures.htm• https://www.coursera.org/course/matrix

The Function The Field The Vector The Vector Space The Matrix The Basis Dimension Gaussian Elimination The Inner Product Orthogonalization

Coding The Matrix

When you take a digital photo with your phone or transform the image in Photoshop, when you play a video game or watch a movie with digital effects, when you do a web search or make a phone call, you are using technologies that build upon linear algebra.

Linear algebra provides concepts that are crucial to many areas of computer science, including graphics, image processing, cryptography, machine learning, computer vision, optimization, graph algorithms, quantum computation, computational biology, information retrieval and web search. Linear algebra in turn is built on two basic elements, the matrix and the vector.

Combining vector addition and scalar multiplicationVector addition = Translation (P’=P+T)

Scalar multiplication = Scaling (P’=SP) or rotating (P’=RP)

Example:

Combining vector addition and scalar multiplication

Matrix-Vector and Vector-Matrix MultiplicationTwo ways to multiply a matrix by a vector:• matrix-vector multiplication• vector-matrix multiplication

• in terms of linear combinations

• in terms of dot-products

For each of these, two equivalent definitions:

Dot-product definition of matrix-vector multiplication: Down Sampling

• Each pixel of the low-res image corresponds to a little grid of pixels of the high-res image.• The intensity value of a low-res pixel is the average of the intensity values of the

corresponding high-res pixels.• Averaging can be expressed as dot-product.• We want to compute a dot-product for each low-res pixel.• Can be expressed as matrix-vector multiplication.

Dot-product definition of matrix-vector multiplication: Blurring

• To blur a face, replace each pixel in face with average of pixel intensities in its neighborhood.

• Average can be expressed as dot-product.• By dot-product definition of matrix-vector

multiplication, can express this image transformation as a matrix-vector product.

• Gaussian blur: a kind of weighted average

Removing perspective

Given an image of awhiteboard, takenfrom an angle...

synthesize an imagefrom straight aheadwith no perspective

Terima Kasih

System of linear equations & Solution setA solution of a linear system is an assignment of values to the variables x1, x2, ..., xn such that each of the equations is satisfied. The set of all possible solutions is called the solution set.

One equation Two equations Three equations

The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3).

The solution set for two equations in three variables is usually a line.

A linear system in three variables determines a collection of planes. The intersection point is the solution.

DeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information about a matrix of coefficients of a system of linear equations, or about a matrix that corresponds to a linear transformation of a vector space.

In the first case the system has a unique solution exactly when the determinant is nonzero; when the determinant is zero there are either no solutions or many solutions. In the second case the transformation has an inverse operation exactly when the determinant is nonzero.

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume (or a higher dimensional analogue) is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.

DeterminantThe area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.

The volume of this Parallelepiped is the absolute value of the determinant of the matrix formed by the rows r1, r2, and r3.

Eigenvalue equationMatrix A acts by stretching the vector x, not changing its direction, so is an eigenvector of A .

Eigenvectors

The transformation matrix preserves the direction of vectors parallel to (in blue) and (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation.Notice that the blue vectors are scaled by a factor of 3. This is their associated eigenvalue. The violet vectors are not scaled, so their eigenvalue is 1

Eigenvalues of geometric transformationsThe following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.

Aljabar Linear• Vectors and Matrices are a staple data structure in many areas of

Computer Science.• Computer Graphics is one prime example—here linear algebra

permeates almost every area.• Basic Linear Algebra — solutions of equations needed in almost every

scientific discipline• Vectors and Matrices — fundamental data structures in computer

science e.g. Arrays, Linked Lists• Numerical Analysis — scientific computing and practical

computational mathematics• Computer Graphics: Transformations, moving object around the

screen, 3D deformations : : : • Image Processing/Computer Vision: Images = matrices, Tracking

objects, Object Recognition, Camera Calibration : : : • Data Compression: JPEG/MPEG, Image/Video/Audio Compression,

Vector Quantisation

Matrices Example: Image Representation

Algebra/Graphs Example: Finite Element Modelling

Matrices Example: Computer Graphics Transformations

Matrices Example: Object Registration/Matching

Matrices Example: ImageWarping (Transformation)

Matrices/Vector Example: Image Compression

Matrices/Vector Example: Image Compression

The three-dimensional Euclidean space R3 is a vector space, and lines and planes passing through the origin are vector subspaces in R3

Mona Lisa eigenvector gridIn this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping, and since its length is unchanged its eigenvalue is 1

Vibration analysisEigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes.

Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, useful for efficient numerical solutions and Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices.

Cremer’s Rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750.

Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is X1 times the first. From this equality Cramer's rule follows.