Q: Show that if A is any m × n matrix, then Im A = A and AIn = A.
A: We know that if A is a matrix of order m x n and B is another matrix of order n x p, then their…
Q: A matrix Q ∈Mn×n(R) is called orthogonal if QQt= I. Prove that if Q is orthogonal, then det(Q) = ±1.
A: The determinant of the product of the matrices is the product of the determinant of the matrices.
Q: Verify that: : is an orthogonal matrix.
A: Given matrix, P=23-23132313-23132323
Q: if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))
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Q: Show that if A is any mxn matrix, then Im= A.
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Q: Prove that the following matrix is orthogonal and skew-symmetric: 1 -1 -1 1 1 M = V3 1 1 1 1.
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Q: 1 0 2 -1 0 2 0 1
A: Given the matrix A=1020-10201 To find an orthogonal matrix P and a diagonal matrix D such that…
Q: Prove that if A is an n × n matrix, then A − AT is skew-symmetric.
A: Let A be an n × n matrix, To prove : A-AT is skew-symmetric, that is (A-AT)T=-(A − AT)
Q: Prove that if A is an n X n matrix, then A - Ar is skew-symmetric
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Q: Let C be a nonsymmetric n × n matrix. For each ofthe following, determine whether the given…
A: Given B=C-CT Find transpose.
Q: Let = Find a nonsingular matrix P and a diagonal matrix D such that P-'AP = D.
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Q: Given 2 B, = { B, =- M. = 0 %3D | 0 Find a) The transition matrix between the ordered bases B1 and…
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Q: In Exercise, compute (a) ||A||2 and (b) cond2 (A) for the indicated matrix.
A: Given matrix is, A=10.911. We have to find the singular values of A. First find ATA as follows:…
Q: Given the bases B={u1 , u2} and B' = {u'1, u'2} for R2, where , u'2 , U2 and u'1= The transition…
A: To find transition matrix find the coordinate of u1 and u2 with respect to basis B'.
Q: Given the bases B={u , Uz} and B' ={u'ı, u'2} for R?, where 1 Uz= 1 and u' u' 2 %3D The transition…
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Q: Given B1 and B2 as the bases of R?. B, = {[",]-C1} and B, = {G-C) %3D Find a. The coordinates of…
A: Ansa; Consider, 10=a11+b3-1⇒10=a+3ba-b⇒a-b=0⇒a+3b=0slove both equation and get;a=b=14 Co-ordinates…
Q: Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗A is a diagonal…
A: Given that A is an m*n matrix in which no two column are identical.Since A*A is defined only if A…
Q: 4. Decide if there is a three by three matrix A such that A. 1 for all æ, y, z in R.
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Q: Let the rows of A ∈ Mn×n(F) be a1, a2, . . . , an, and let B be the matrix in which the rows are an,…
A: This can be done as follows:
Q: 1. Find a condition on a, b, c, d such that the augmented matrix " E b has a unique solution.
A: The condition on a,b,c,d must be a relation between a,b,c,d.
Q: Given the bases B={u , u2} and B' ={u'1, u'2} for R2, where U2 and u'1- u' 2 The transition matrix…
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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: b) Prove: Let A be an m × n matrix, R" = NS(A) RS(A).
A: Given: Let A be an m×n matrix , Rn=NSA⊕RSA
Q: If A is an n X n matrix, prove that det(adj A) = (det A)n-1
A: Given the matrix A of order n×n, and we have to prove that det(adjA) = (detA)n-1
Q: Show that if A is any m × n matrix, then AAT and AT A are symmetric.
A: There is a matrix A given of order m×n. To show that AAT and ATA are symmetric. Solution: The…
Q: Show that if A is an orthogonal matrix, then det(A) = ±1.
A: Given If A is an orthogonal matrix. To show det(A) = ±1
Q: If A is a skew-symmetric n X n matrix and n is odd, prove that det A = 0.
A: Since A is a skew-symmetric matrix we have, AT=-A where AT is the transpose of matrix A. Then we…
Q: Let A be the following matrix 6 7 4 9 5 9. 3 4 After diagonalization, P and Dare found such that A =…
A: Use the following theorem: If A and B are two n by n matrices, then trace of (AB)=trace of (BA)
Q: Let Q be an orthogonal matrix and let d = det(Q). Show that |d| = 1.
A: Given that Q is an orthogonal matrix, which implies that
Q: 4. Given B1 and B2 as the bases of R?. B, = {[).} {HE) and B2 = · Find a. The coordinates of…
A: If a vector is written as, a= c1v1+c2v2, then c1,c2 are the coordinate of that vector a. Then method…
Q: If A is a nonsingular matrix such that A2 = A, what is det(A)?
A: Given: A is a non-singular matrix such that A2=A. To find: A Solution: Non-singular matrix is a…
Q: Given the bases B={u , U2} and B' ={u'1, u'2} for R2, where and u'1= u' 2 U2 from B to B' is: The…
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Q: For a matrix A having JA| = 0, which of the following are true? * O A- exists with |A¬' |= 0| O A-…
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Q: Given the bases B={u , U2} and B' = {u'1, u'2} for R2, where U2 and |=ן 'u u'2 The transition matrix…
A: Transition matix is using for change of basis. Here we are changing basis B to B'.
Q: Calculate adj A, det A, and hence A¯', for the matrix A=| 2 1 -2 -1 1
A: To find the adjoint determinant and inverse of a matrix
Q: Show that if A is an orthogonal matrix, then det(A) = +1.
A: To prove, detA=±1
Q: 3. Let Px and Pw be perpendicular projection matrices with C(Pw) CC(Px). Show that Px - Pw is a…
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Q: Suppose that A is an n × n matrix and det A + 0. Determine the rank and nullity of A.
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Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…
A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…
Q: Let B = {uj = ,U2 = and C = V2 = Find Pg, the transition matrix from the standard basis of R? to the…
A: Given B = u1=21 , u2=11 We have to find PB , the transition matrix from the standard basis of ℝ2…
Q: 1. Find a matrix P, that orthogonally diagonalizes A,and determine P-1 AP 3. A = %3D 1
A: First we calculate eigenvalues and eigenvectors of matrix A
Q: Given the bases B={u , u2} and B' ={u'1, u'2} for R2, where • U2 and u'1= u'2 = The transition…
A: Let B1 ={u1,u2.....un} be a basis of n- dimensional vector space V and let B2 ={v1,v2....vn} be…
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 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: et A be the following matrix 2 7 2 4 4 8 2 3 3 fter diagonalization, Pand Dare found such that A =…
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Q: show that if At exists for matrix A > then A is to
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Q: Let B = , U2 = and C = V1 = Find Pe, the transition matrix V2 = from the standard basis of R? to the…
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Q: and D be a diagonal matrix D = diag(A₁,..., An) such that
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Q: Prove that if A is an orthogonal matrix, then so are AT and A−1.
A: We have to prove that if A is an orthogonal matrix, then so are AT and A−1. If A is an orthogonal…
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- Let a=<−5,0,−3> and b=<4,0,2> Show that there are scalars s and t so that sa+tb=<27,0,15>.s = ? t = ?A) compute x the component of x that is parallel to L and x the component of x that is orthogonal to L B) compute the reflection of x across the line L; refL(x)Suppose (B - C) D = 0, where B & C are m x n matricies and D is invertible. Show that B = C.
- Find the moving trihedral of C for all t ∈ (0, π). [ THIS IS NOT A GRADED QUESTION ]4. Draw a set of orthogonal 2d axes and illustrate the play that intersects at X=1/4, Y= 1 and Z=1/2 Calculate the Miller indices(a) Prove that u + vand u - v are orthogonal in IR" if and only if ||u|| = ||v||. (b) Draw a diagram showing u, v, u + v, and u - v in IR2 and use (a) to deduce a result about parallelograms.
- The orthogonal projection of v = 10i on b = −3i + j is _____ .In C[0,1] with inner product defined by <f,g>=integral from 0 to 1 of f(x)g(x)dx, consider vectors 1 and x. (a) find the angle theta between 1 and x (b) determine the vector projection p of 1 onto x (c) verify that 1-p is orthoganal to p (d) compute ||1-p||, ||p||, ||1|| and verify that the pythagorean law is satisfiedIf u = (3, −1, 4) and v = (−2, 3, 5), the scalar projection of y onto v is:
- Let y = [ 2 3 -1] and u = [ 2 -6 -6]compute the distance d from y to the line through u and the originIn Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, . . . , xn)T and y = (y1, . . . , yn)T .Let A = (1,1) and B = (1,5) be in the Poincar ́e plane. Find C = (xC,yC) for which xC > 1 and mH(∠ABC) = 135.