Examples of Group Actions
... F → S → G\S of the inclusion of F ,→ S and the canonical projection S → G\S is a bijection; we say that F is a fundamental domain if this is so. Also, recall that for each x ∈ S, there is a natural bijection from the coset G/StabG (x) to the orbit G · x. 1. Let G = µ2 = {1, −1}, S = R, and the actio ...
... F → S → G\S of the inclusion of F ,→ S and the canonical projection S → G\S is a bijection; we say that F is a fundamental domain if this is so. Also, recall that for each x ∈ S, there is a natural bijection from the coset G/StabG (x) to the orbit G · x. 1. Let G = µ2 = {1, −1}, S = R, and the actio ...
A Tutorial on MATLAB Objective: To generate arrays in MATLAB
... 4.2 Use MATLAB, to plot different functions given below in one figure for each sections a, b, c, and d. a. b. c. d. 5. Objective: To create matrices by entering row/column elements and by forming from vectors. Procedure: The most direct way to create a matrix is to type the matrix row by row, separa ...
... 4.2 Use MATLAB, to plot different functions given below in one figure for each sections a, b, c, and d. a. b. c. d. 5. Objective: To create matrices by entering row/column elements and by forming from vectors. Procedure: The most direct way to create a matrix is to type the matrix row by row, separa ...
Page 1 Solutions to Section 1.2 Homework Problems S. F.
... A general solution of a system is an explicit description of all solutions of the system. True. See page 21 of the textbook. 23. Suppose a 3 5 coefficient matrix of a linear system has three pivot columns. Is the system consistent? Why or why not? The system is consistent because each row of the ...
... A general solution of a system is an explicit description of all solutions of the system. True. See page 21 of the textbook. 23. Suppose a 3 5 coefficient matrix of a linear system has three pivot columns. Is the system consistent? Why or why not? The system is consistent because each row of the ...
Math 611 HW 4: Due Tuesday, April 6th 1. Let n be a positive integer
... with 1 ≤ i, j ≤ n, define the matrix Xij (α) to be the matrix with xkk = 1 for all k, 1 ≤ k ≤ n, xij = α, and all other entries are zero. Prove that Xij (α) has determinant one. 2. If n is a positive integer and F is a field, show that SL(n, F) is a normal subgroup of GL(n, F). Also, if F has q elem ...
... with 1 ≤ i, j ≤ n, define the matrix Xij (α) to be the matrix with xkk = 1 for all k, 1 ≤ k ≤ n, xij = α, and all other entries are zero. Prove that Xij (α) has determinant one. 2. If n is a positive integer and F is a field, show that SL(n, F) is a normal subgroup of GL(n, F). Also, if F has q elem ...
Sept. 3, 2013 Math 3312 sec 003 Fall 2013
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
18.03 LA.2: Matrix multiplication, rank, solving linear systems
... It is helpful to see the geometric picture in these small cases, because the goal of linear algebra is to deal with very large matrices, like 1000 by 100. (That’s actually a pretty small matrix.) What would the picture of a 1000 by 100 matrix be? It lives in 1000-dimensional space. What would the 10 ...
... It is helpful to see the geometric picture in these small cases, because the goal of linear algebra is to deal with very large matrices, like 1000 by 100. (That’s actually a pretty small matrix.) What would the picture of a 1000 by 100 matrix be? It lives in 1000-dimensional space. What would the 10 ...
Course Code
... Mathematics is the language of universe. For this reason it is vitally important to find and apply the mathematical equivalent of any problems that emerge in our daily lives. Therefore, the course aims at giving the basic mathematics and providing the business administration students the capacity to ...
... Mathematics is the language of universe. For this reason it is vitally important to find and apply the mathematical equivalent of any problems that emerge in our daily lives. Therefore, the course aims at giving the basic mathematics and providing the business administration students the capacity to ...
The Fundamental Theorem of Linear Algebra Gilbert Strang The
... The column space is the range R(A), a subspace of Rm. This abstraction, from entries in A or x or b to the picture based on subspaces, is absolutely essential. Note how subspaces enter for a purpose. We could invent vector spaces and construct bases at random. That misses the purpose. Virtually all ...
... The column space is the range R(A), a subspace of Rm. This abstraction, from entries in A or x or b to the picture based on subspaces, is absolutely essential. Note how subspaces enter for a purpose. We could invent vector spaces and construct bases at random. That misses the purpose. Virtually all ...
Matrices - MathWorks
... In general, determinants are not very useful in practical computation because they have atrocious scaling properties. But 2-by-2 determinants can be useful in understanding simple matrix properties. If the determinant of a matrix is positive, then multiplication by that matrix preserves left- or rig ...
... In general, determinants are not very useful in practical computation because they have atrocious scaling properties. But 2-by-2 determinants can be useful in understanding simple matrix properties. If the determinant of a matrix is positive, then multiplication by that matrix preserves left- or rig ...
Definitions:
... mathematicians have given it a special name: the eigenvalue/eigenvector problem. This problem can be solved like such: ...
... mathematicians have given it a special name: the eigenvalue/eigenvector problem. This problem can be solved like such: ...
31GraphsDigraphsADT
... Relationship between the Adjacency matrix and the Path matrix for a digraph Suppose A is the adjacency matrix. Let matrix B = Ak Then bij is the total number of distinct sequences < n1, . .> . . . . . . <. . , nj > that: i) have length k ii) correspond to paths in the digraph Proof: For k = 1 then B ...
... Relationship between the Adjacency matrix and the Path matrix for a digraph Suppose A is the adjacency matrix. Let matrix B = Ak Then bij is the total number of distinct sequences < n1, . .> . . . . . . <. . , nj > that: i) have length k ii) correspond to paths in the digraph Proof: For k = 1 then B ...