Let A be the matrix of the linear transformation T- Without writing A-.docx

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Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace. T is the transformation on R3 that rotates points about some line through the origin. Solution T is the transformation on R 3 that rotates points about some line through origin.Let the line be fixed. Let (x,y,z) be a point on the line. Then any other point on it is c(x,y,z) for some real number c. Now t is an eigen value of T means, there is a non zero (a,b,c) such that (T-tI)(a,b,c)=0. where I is the identity transformation. That is T(a,b,c)=t(a,b,c) So we are looking for scalars t and vectors (a,b,c) such that above equation holds. To be more precise, we want that (a,b,c) which are just multiplied by a constant t if T is applied on them. In other words, we are looking (a,b,c) such that this point is just fixed or moved along some line through origin origin. Since T is a rotation, the only points that are not moved are the points on the line about which the rotation takes place. Also the points on the line are fixed. That is T(x,y,z)=(x,y,z) for all points (x,y,z) in the line. So 1 is an eigen value of T and the given line(since it is not moved by T) is the eigen space corresponding to 1. .

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Hi please complete the following with detailed working out Find the eigenvalues of C1 C1 C1 C1 C2 C3 C3 C3 C3 cn cn Hint: You may find that the identity A2 (c1 c2 cm) is useful. Solution COORDINATES, EIGENVALUES, AND EIGENVECTORS 1. Review of coordinates 1.1. The standard basis. The standard basis for R n is the basis E = {e1, e2, . . . en}, where ej is the vector with 1 in the j-th position and zeros elsewhere. 1.2. Coordinates with respect to a basis. Let B = {v1, v2, . . . vn} be a basis for R n . We know that for each vector x R n there are unique scalar “coordinates” c1, c2, . . . cn such that (1.1) x = c1v1 + c2v2 + . . . + cnvn We call the vector c R n having coordinates c1, c2 . . . cn the “coordinate vector” of x with respect to the basis B, and denote it by [x]B. 1.3. First examples. (a) [x]E = x for each x R n . (b) For any basis B = {v1, v2, . . . vn} of R n : [vj ]B = ej (1 j n). (c) [0]B = 0 for any basis B (the zero vector has the same representation with respect to any basis. 1.4. Matrix calculation of coordinates. Letting B = {v1, v2, . . . vn} be a basis for R n we can rewrite equation (1.1) in matrix form as (1.2) x = V [x]B where V is the matrix having the vector vj for its j-th column, i.e. V = [v1, v2, . . . vn]. . 1 2 EIGENVALUES/EIGENVECTORS V is invertible because its columns are linearly independent (see, for example, Summary 3.2.9(v) of our textbook, pp. 119–120), so we can compute the B coordinate representation of any vector x R n by solving (1.2) for [x]B: (1.3) [x]B = V 1x For example, if in R 2 we take B = {v1, v2} to be our basis, where v1 = 1 3 and v2 = 2 5 , then V = 1 2 3 5 and V 1 = 5 2 3 1 . Suppose, for instance, that x = 1 1 . Then [x]B = V 1x = 5 2 3 1 1 1 = 7 4 More generally, if x = x1 x2 is any vector in R n then the same calculation shows that [x]B = 5x1 + 2x2 3x1 x2 . 1.5. The matrix of a linear transformation with respect to a basis. Recall that each n × n matrix induces a linear transformation T : R n R n by means of matrix multiplication: (1.4) T(x) = Ax (x R n ). We refer to A as the “standard matrix” for T. The lesson of what’s to follow is that it’s not always the best matrix for T! For a basis B = {v1, v2, . . . vn} for R n we define (1.5) [T]B = [[T(v1)]B, [T(v2)]B . . . [T(vn)]B] , i.e. [T]B is the matrix that has the B-coordinate representation of T(vj ) as its j-th column (1 j n). We call [T]B the matrix of T with respect to the basis B. 1.6. Example: B = E, the standard basis. In this case [T]B = [Ae1, Ae2, . . . Aen], the “standard matrix” for T. EIGENVALUES/EIGENVECTORS 3 1.7. Geometric examples in R 2 . In this next group of examples, L is a line through the origin in the direction of a non-zero vector v R 2 , and w is a non-zero vector perpendicular to v. We take as our basis for R 2 , not the standard one, but instead B = {v, w}. (a) Suppose T : R 2 R 2 is the transformation of orthogonal projection onto L. Then T(v) = v and T(w) = 0, so [T]B = [[T(v)]B, [T(w)]B] = [[v]B, [0]B] = .

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*This is a question from Mathmatical Inqiury class. In which you have to use logic to solve these problems. A complete 4-partite graph is a graph in which the vertices (points) are partitioned into 4 sets. There are no edges among the vertices of a given partite set, but every vertex in a set is connect by an edge to every vertex in the 3 sets in which it\'s not contined. How many vertices are in the graph K_r_,_s_,_t_,_u_ ? How many edges does the graph contain? The sign _s_ indicates subscript s. Solution The first question, the number of vertices, is easy. The notation for a 4-partite graph, Kr,s,t,u, indicates that the 4 sets of vertices the graph is partitioned into (as you defined in the problem) have r, s, t, u vertices in them respectively. The definition of a partition of S is a set of disjoint sets whose union is S. Since the four sets are a partition, they are disjoint, meaning no vertex appears in more than one of them, and their union is V(Kr,s,t,u), meaning that every vertex must appear in one of the four sets. Therefore, by counting the vertices in each partition and adding them, we get the number of vertices in the graph, since we are counting each vertex exactly once -- counting one twice would mean it appeared in more than one set and missing a vertex would mean that vertex isn\'t in the union of the sets, either of which contradicts the sets being a partition. And for the number of vertices we therefore have the sum of the numbers of elements in each of the four partitions: Number of vertices = r + s + t + u The number of edges is a bit harder. Call R the set of vertices with r elements, S the one with s elements, T the one with T elements, and U the one with u elements. Choose an arbitrary vertex x in R. Then, since the graph is a complete 4-partite graph, we have an edge that connects x to every vertex in S (s vertices, which means s edges connecting x to the vertices in s), an edge that connects x to every vertex in T (t vertices, which means t edges needed to connect to them), and an edge that connects x to every vertex in U (u vertices, which again means u edges from x to a vertex in U). So every vertex x in R has s+t+u edges incident to it. The total number of edges incident to all the vertices in R combined is the number of vertices in R times the number of edges incident to each of them, or r(s + t + u) Symmetrically, the number of edges incident to all the vertices in S put together is s(r + t + u), the number of edges incident to all the vertices in T put together is t(r + s + u), and the number of edges incident to all the vertices in U put together is u(r + s + t). Now, since we have examined every vertex in R, S, T, and U, and we know that every vertex is in one of these sets (again, becaues they form a partition), it might be logical to think that the total number of edges is the sum of the numbers of edges that are incident to the vertices in each set: r(s + t + u) + s(r + t + u) + t(r + s + u) + u(r + s + t).

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Light travelling through the air (n = 1) strikes the surface of a diamond (n = 2.42) at a large angle of incidence. Which of the following describes the light ray as it enters the diamond? a) the light bends toward the normal to the surface, and slows down b) the light bends toward the normal to the surface, and speeds up c) the light bends away from the normal to the surface, and slows down d) the light bends away from the normal to the surface, and speeds up e) the light ray does not enter the diamond, but totally reflects and stays in the air Solution When light is enter into rare medium to denser medium than light rays bends towards the normal. Light slows as it travels through a substance and every substance has an index of refraction which defined as the ratio of the speed of light in vacuum to the speed of light in that substance. The index of refraction for diamond is 2.4 which means that light is 2.4 times slower in diamond . Hence Option A is correct. Hope it will help you. Please rate the answer. .

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Let L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7: Is L a linear map? If it is, find as many independent vectors in Ker(L) as you can. Solution L(x1; x2; x3; x4; x5; x6; x7) = 2x3 + 3x4 + 8x6 + x7 Since L satisfies scalar multipication and addition properties, it is a LINEAR map rank of transformation matrix = 4 =>dimension of kernal = 7-4 = 3 =>maximum number of independent vectors in kernel = 3 =>these vectors are : (1 0 0 0 0 0 0), ( 0 1 0 0 0 0 0) , (0 0 0 0 1 0 0 ) .

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list examples of the internet entities( switches, web servers, mail server, users and system) need to communicate securely? Solution ARPANET was designed to support the interconnection of networks ( inter-networking ). Routers connect the various individual networks and links together that make up the Internet. switch is a computer networking device that connects devices together on a computer network, firewall is a combination of hardware and software that isolates an organizationâ€™s internal network from the Internet at large, allowing some packets to pass and block- ing others. A web server is an information technology that processes requests via HTTP, the basic network protocol used to distribute information on the World Wide Web. users navigate from one web page to another via hyperlinks embedded in the documents. These documents may also contain any combination of computer data, 1) User end system (customer) and web server (online store) in business transactions 2) User end system (laptop) and a password protected router (Wi-Fi internet) during log on 3) Two web servers (email service) in sending an email .

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Linux distributions are becoming more and more user friendly. There is a relatively new distribution called Ubuntu that is extremely easy to install and is very functional. What additional Linux distributions are currently available, and what are their advantages and/or disadvantages? Solution The various linux distributions available are given below: 1) Finnix, its used for system administration and also comes in market with various architectures 2) Raspbian, its a a type of desktop bases distribution, its also known as a very less power computer 3) 64 Studio, its for the production of audio and video on x86 environment. 4) Corel Linux, its also one of the linux distribution. 5) Dream Linux, its one of the old linux distribution Advantages of Linux Distributions: 1) One of the advantages is that, most of the distributions of the linux are free to use and it also can be installed without license in any numbers of Computers. 2) They are more secure than windows systems, with less virus attacks 3) Opne sources and you can free to do the modifications on it. 4) Various software available to use on these distributions. 5) Robust and fast in execution. Disadvantages of Linux Distributions: 1) To understand linux distribution, you have to be patience, as its not easy as windows, to get familiar with it. 2) Not have a very strong GUI as windows. 3) Linux distributions also having various compatibility issues with other systems. 4) In terms of various applications, linux distributions are still lagging behind the windows .

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