PPT-8.6.2 – Orthogonal Vectors

At the end of yesterday we addressed the case of using the dot product to determine the angles between vectors Similar to equations from algebra we can talk about relationship of vectors as well Parallel. 1 Motivation A3 A2 Vectors A3 A21 Notational Conventions A4 A22 Visualization A5 A23 Special Vectors A5 A3 Vector Operations A5 A31 Transposition A6 A32 Equality A6 A33 Addition and Subtraction Orthogonal matrices. independent basis, orthogonal basis, orthonormal vectors, normalization. Put orthonormal vectors into a matrix. Generally rectangular matrix – matrix with orhonormal columns. Square matrix with orthonormal colums – . Five-Minute Check (over Lesson 8-4). Then/Now. New Vocabulary. Key Concept: Dot Product and Orthogonal Vectors in Space. Example 1: Find the Dot Product to Determine Orthogonal Vectors in Space. Example 2: Angle Between Two Vectors in Space. . can. be . interpreted. as a file of data. A . matrix. . is. a . collection. of . vectors. and . can. be . interpreted. as a data . base. The. red . matrix. . contain. . three. . column. Fundamental system in linear algebra : system of linear equations . A. x . = . b. . nice case – . n. equations, . n. unknowns. matrix notation. row picture. column picture. linear combinations. For our matrix, can I solve . A.D. . Rollett. Vectors, Matrices, . Rotations. , . Axis Transformations. Most of the material in these slides originated in lecture notes by Prof. Brent Adams (. now emeritus . at BYU). . Last revised: 9 Nov. ‘11. Hung-yi Lee. Outline. Reference: Chapter 7.1. Norm & Distance. Norm. : Norm of vector v is the length of v. Denoted . Distance. : The distance between two vectors u and v is defined by .  .  .  . In some cases, we will have to decompose a vector into a sum of two separate vectors. Recall; most vectors may be written as some variation of the special unit vectors {1,0} and {0,1}. With vectors, sometimes they may not be pointing or oriented in the proper direction. Division . Multiplexing ). Basics of . ofdm. Orthogonal Frequency Division Multiplexing.  (OFDM) is a method that allows to transmit high data rates over extremely hostile channels at a comparable low complexity.  . Any vector can be resized by multiplying it by a real number (scalar).. Multiplying by positive scalar changes magnitude only.. Multiplying by a negative scalar changes the magnitude and its direction.. Lijie. Chen. MIT. Ryan Williams. MIT. Fine-Grained Complexity:. “Modern” NP-completeness. Many Conceptual Similarities. NP-Completeness. Which problems require . super-poly time. ?. Fine-Grained Complexity. اعداد ا.م.د. . عبدالخضر. اللامي. A.G.Farhan. . #The dot product of u=(u. 1. ,…,u. n. ) and . v=(v. 1. ,…,v. n. ) is written u.v and is defined two ways:. 1. u.v= u. 1. The characteristic roots of the (. p×p. ) matrix . A. are the solutions of the following determinant equation: . Laplace expansion. is used to write the characteristic polynomial as. :. Since (. A vector is a DNA molecule that has the abil­ity to replicate autonomously in an appropri­ate host cell and into which the gene of inter­est (a foreign genetic sequence) is integrated. When we insert a foreign genetic sequence into the vector the aim is either to obtain numer­ous copies of the gene of interest or to obtain the product of that..