5. Let S be the set of all points (x11, X12, X13, X22, x23; X33) E K® such that the matrix x11 *12 x13 *12 T22 X23 T13 223 r33 is positive semidefinite. Is the set S convex? Justify your answer.

Q: Let A be an n × n matrix with triangular factorization LU. Show that det(A) = u11u22 · · · unn

A:

Q: Let matrix A be p x q.  Prove that ATA is singular if the rank of A is strictly less than q.

A: Given A=aijp×q Suppose rankA<q Now, ATA is a matrix of order q×q. Now, ATA is non-singular if and…

Q: In Exercises 9–10, find bases s for the null space and row space of 2. b. А —D | 4 -1 3 -1] 9. а. А…

A: Here in this question we need to find out the basis for null space and row space of the given…

Q: Prove that if A is an m x n matrix of rank r, then the dimension of the solution space of Ax = 0 is…

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Q: 2. Show that if A is an n x n real symmetric matrix then (Ax) y = x (Ay) for all x, y E R".

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Q: 3. Let A be a m X n matrix with columns C1, ... , Cn . Prove that {c1, ..., C,} is independent if…

A: In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a…

Q: A. Let R be a relation on set A. A={x :xeJ and 0 sx s 12), and R={(a,b) : la - b| is multiple of 4}.…

A: Since not particular question asked, as per guidelines solution to only first question(i.e A) is…

Q: Let K be an nxn matrix over Z26 such that K = K-1. Explain why either det(K) = 1 or %3D det (K) =…

A:

Q: How many rows and columns must a matrix A have in order to define a mapping from R4 to R6 by the…

A: Given: A mapping from R4 to R6 by T(x)=Ax Find how many rows and columns having the matrix A

Q: 5. Compute the condition number K(A) of the following matrix 1 1+€ 1 1 1 A 1 1 1 1+€ What is k(A) as…

A: Conditional number is κ∞(A)=∥A∥∞·∥A-1∥∞.If matrix A is known to us so ∥A∥∞ is easy to find .First…

Q: r every square matrix A = (a;;) € Μπ(C) η |4||, = sup ||Ax|t Σ = max Elai TEC ||z||1=1 π Alloo sup…

A:

Q: 5. Let D be a non-singular matrix such that 7. (7D)- 1 -2 Find D.

A: Inverse matrices

Q: Prove that k(x,x)=xAx' is a valid kernel, where A is a symmetric positive semidefinite matrix.

A: Let  we consider      kx,x' = xTAx'…

Q: 8. Let T-(::) a c d be a 2x2 matrix with real valued entries. Prove that there exists M ER such that…

A: We use some useful inequalities.

Q: 4. Prove that if Q is a Householder matrix corresponding to the unit vecto u, then r v ifv is in…

A: Given that, Q is a Householder matrix corresponding to the unit vector u. The householder matrix…

Q: How many rows and columns must a matrix A have in order to define a mapping from R* into R³ by the…

A: Find the number of Rows and columns in a matrix A.

Q: Suppose Q(x) = x" Ax %D for some symmetric matrix А, and that we have a diagonal matrix D and a…

A: We know,  A symmetric matrix is a square matrix that is equal to its transpose. Any square matrix…

Q: If Q+A= A holds for every m×n matrix A, show that Q = 0mn.

A: Consider the given equation, Q=A+A   Now, subtract the matrix A from both side: Q+A-A = A -A As, A…

Q: Which of the following are true for the adjacency matrix of the attached graph? 1. ||A||F = 4 2.…

A: Given: A graph To find: Which of the following are true for the adjacency matrix of the attached…

Q: Findthe Linear Eransfromation whose matrix MT with regard to the base to mec2And the natural ruleto…

A: Given T:C2→C3 be a linear transformation whose corresponding matrix is given  MT=i-i1+i-21-1+2i and…

Q: 11. Let A be an m x n matrix and let x, y e R". Prove that if Ax = 0 and y = ATb for some b e R",…

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Q: 13. Find an estimate for det (A) if A is an n × n matrix whose entries form some pattern of +10's

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Q: The matrix that implements the mapping T (₁, 2) = (-3x1 +4x2,0, 7x2 - 5x1,1 + 2x₂) is -3071 4 052 0…

A:

Q: (c) 0 is neither diagonalizable nor

A:

Q: 31. Let A be a 5 ×7 matrix with rank 4. a. What is the dimension of the solution space of Ax = 0? b.…

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Q: A matrix B in Mnxn (C) is called skew-symmetric if Bt= -B. Prove that if B ϵ Mnxn (C) is…

A: Let B be skew-symmetric matrix of order n, where n is odd, then we have

Q: For every matrix A, it is true that (AT)T = A O True O False

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Q: §8.3, Exercise 5. Suppose the mapping L : R³ → R² is linear and satisfies (:)-(3) ()-6) ()- ) 1 1 2…

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Q: Let A = {p, q, r, s} and the relation R on A is represented by the matrix 0 0 1 1 c) 0 1 0 0 MR .…

A: As per the guidelines, I am supposed to do the first problem only. Kindly pst the other question…

Q: Compute the standard matrix for the linear map T : R² → R² which (1) rotates i by 0 = T, (2)…

A:

Q: Construct the dual of the following Convex optimization Hinimize IJ Ay-bu Subject to Qx =P full rank…

A: Given, Minimize Ax-b22Subject to,                   Qx=P And A and Q are m×n and p×n full rank…

Q: :Prove that the function f(x₁, x,)=- X₂ X₂ purpose, compute the Hessian matrix for f(x). defined on…

A: The given function is f(x1,x2)=1x1x2 defined on the domain ℝ++2. The objective is to prove that…

Q: O Give the matix set and condiition up Matrix. of @Optimize which Boarded Hessian 2 2x²4x 2.…

A: minimum value of the function

Q: Let A be a nonsingular k x k matrix with k> 1. Show that. det [adj(A)] = [det(A)]*-1

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Q: Let Mı5(IR) = {A/A and 15th order square matrix with real elements} and consider the linear…

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Q: Show that 3 10 S = 1 10 is a linearly dependent set in R³.

A: Given set of vectors in R^3 are: {(2, 1, 3), (3, -1, 2), (10, 0, 10)} We know that three vectors u,…

Q: Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in a line and connected by…

A: (a) The 4 x 4 matrix equation relating the column vectors f and x will be:

Q: Let A be an m x n matrix. Is the set K defined below convex ? Why or why not ? K = {y € R" : 3r E R"…

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Q: Let A 2 3 2. 4 o 1 11 2 3 5 36 A has pivots. rankAl = =D The dimension of the null space of A is [

A: GivenA=1023240101123536To find1)Number of pivot  of A2Rank(A)3)Dimension of null space of A

Q: Let M € M₂ (F), A € M₂(F), C = M(F) and B € Mpxk (F) with n=p+k. Also let O denote the kx p zero…

A: Given M∈Mn(F), A∈Mp(F), B∈Mp×k M=ABOIk Here we have M=Ap×pBp×kOk×pIk It means Above matrix is square…

Q: How many rows and columns must a matrix A have in order to define a mapping from R into R by the…

A:

Q: A matrix Q ϵ Mnxn(R ) is called orthogonal if QQt = I . Prove that if Q is orthogonal, then det(Q) =…

A: Given data: The orthogonal matrix is given by "Q". The given condition is  QQT = I.....(1) Now, take…

Q: Suppose V and W are finite-dimensional and T E L(V,W). Show that with respect to each choice of…

A: Suppose V and W are finite-dimensional and T∈LV,W. To show: to each choice of bases of V and W, the…

Q: Find the Jordan canonical form J for -10 -2 2 B = -4 2 -1 -4 2 4 -1 1 3 Also, find a matrix Psuch…

A: Given :  B = -10-2-421-x24-42-1-x-42-113-x ⇒p(x)=det(A-xI)=-1-x0-2-4x1-x24-42-1-x-42-113-x p(x)=…

Q: Suppose that A is a 3 x 12 matrix, and that 7(x) = Ax. If T is onto, then what is the dimension of…

A: We know that the null space of a matrix A of order n×m is the number of linearly independent…

Q: Let A be a 4x4 singular matrix. What is the smallest possible dimension of Null Space of A? A) 2

A: Using,  Rank- Nullity Thereom

Q: Let X be a centered m x n-matrix. Then cov(X) = XTX. %3D Select one: O True O False

A: It is an important part of statistics . It is widely used .