20. Let V be a finite-dimensional inner product space and let U be a linear operator on V. Then U is unitary if and only if the matrix of U in some (or every) ordered orthonormal basis is a unitary matrix.

Q: If A be an n x n invertible matrix. Prove that the transpose of A is also invertible.

A: A square matrix is invertible matrix if its determinant is nonzero. To obtain the transpose of the…

Q: Let {u1, u2, u3} be an orthonormal basis for an innerproduct space V and letu = u1 + 2u2 + 2u3 and v…

A:

Q: Let M2x2 be the vector space of all 2 x 2 matrices, and define -[: :} T: M2x2 → M2x2 by T(A) = A +…

A:

Q: (12) Let T be a diagonalizable linear operator on V-finite dimensional, and let m be any positive…

A:

Q: Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

A:

Q: 2. Prove that a 2 x 2 matrix A with entries in Z26 is invertible if and only if gcd(det(A), 26) = 1.

A:

Q: Prove that the spectral norm of a symmetric matrix is the same as its spectral radius.

A:

Q: 14. Let T be a linear operator on a finite-dimensional inner product space V. Then T is self-adjoint…

A:

Q: 4. Verify that M, = 1- Px = 1- X(X'X)-X' is symmetric and idempotent for any matrix X, where I is a…

A: A matrix A is symmetric if

Q: 2. Let A be an involutory matrix. Prove that hat1 + A) is an idempotent matrix,

A:

Q: Show that if A and B are invertible matrices, then cond(AB) <= cond(A) cond(B) with respect to…

A:

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: Let U be the vector space of 3×2 matrices over R. Show that dim.U =6.

A:

Q: 3. Show that the hat matrix is symmetric and idempotent. (That is, show that H' = H and that НН -…

A:

Q: 1 . Let V be a finite dimensional inner product space. The thogonal if and only if its matrixr…

A:

Q: 6. Let V be the vector space of all diagonal 5 x 5 matrices, then dim(V)=--

A: Given that the vector space contains all diagonal 5×5 matrices and we need to find its dimension.

Q: Please see image

A: Hi! This is an Advanced Math question. We are answering it now but kindly post such questions under…

Q: Show that if A is a reversible matrix on an F body, the At transpose is also reversible.

A: Given that A is a reversible matrix. Then, AA-1=I, where I is an identity matrix and A-1 is the…

Q: Let O(2, R) be the set of 2 x 2 orthogonal matrices. Describe all of the O(2, R)-conjugacy classes.

A: (1) Two elements x and y of a group G are said to be conjugates if there exists an element g in the…

Q: tet A be an mxn matrix Prove that on

A: Let A be an m×n matrix prove that Frobenis norm : AF=∑i=1m∑j=1naij212↔·2 on Cmn

Q: Let V = {Set of all invertible N × N matrices} with the standard definition of matrix addition and…

A: A set of vectors V defined on a field F with two operations  vector addition and scalar…

Q: 20) Let W be the vector space of all symmetric 2 x 2 matrices. Show that W sisomorphic to R³

A: Introduction: The set of vectors v1, v2, ..., vn in V is said to be linearly independent if the only…

Q: Let U be a unitary matrix. Prove that ||Ux|| = ||x|| for all x ∈ Cn.

A:

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: 6. The set of all 2 x 2 matrices of the form a. is a vector space. b. is not a vector space because…

A: The solution are next step ...

Q: 3. Suppose V is a 5-dimensional complex vector space and let T € L(V). A basis for V may be chosen…

A:

Q: b) If the matrix has an A = PDPT, prove that A is a orthogonal diagonalization šymmetric matrix

A: If the matrix has an orthogonal diagonalization A= PDPT, then to show A is a symmetric matrix…

Q: Let A be mxh matrix. an Prove that. AlL %3D max

A: (Solving the one question as per our guidelines) Solution: Given A is an m×n matrix. We have to…

Q: If (e,} is an orthonormal basis for a Halbert space H, prove that || x ||2=£ | |² for all x in H.

A: An inner product space which is complete with respect to the norm coming out of the inner product is…

Q: Suppose there is a basis a of R2, for which the system |[T]-AI|=0 has one, and only one, real…

A: Every linear operators in vector spaces with real scalars and real eigenvalues Every linear…

Q: onsi with its standard product. nat A E Mnxn(R) is orthogonal if and only if, for any orthonormal…

A: since {v1,...,vn} is orthonormal , thenvit * vj=0 if i different than j and 1 if i=j where I wrote…

Q: 5) Give an example of an operator whose matrix with respect to some basis contains only nonzero…

A: We have to give an example of an operator whose matrix with respect to some basis contains only…

Q: Let B₁ and B₂ be orthonormal bases for an n-dimensional vector space V. Let A = JB₂B₁ be the…

A:

Q: Let U be a unitary matrix. Prove that if λ is an eigenvalue of U, then |λ| = 1.

A:

Q: Let V be a finite-dimensional inner product space and let U be a linear operator on V. Then U is…

A:

Q: Let A be an n x n diagonalizable matrix. Suppose T : R" → R" is transformation with the standard…

A: Let A be n×n diagonalisable matrix. Suppose T:Rn→Rn be a linear transformation with the standard…

Q: 6. The set of all 2 × 2 matrices of the form a. is a vector space. b. is not a vector space because…

A:

Q: Prove that the null space of an mxn matrix is a sub space of R^n

A:

Q: An n x n matrix A is invertible if ВА In АВ II

A: Inverse Matrix: Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I = BA,…

Q: (a) Find the coordinate vector of z 3 with respect to the ordered basis of R (b) Let F, be the…

A: Given that x=-5  3-1 is a coordinate vector and ordered basis is E=157,01-2,001.

Q: Let V be the set of all 2 × 3 matrices. Verify that V is a vector space.

A:

Q: Let U be a unitary matrix. Prove that (a) U is also a unitary matrix.

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Is this statement true or false? EXPLAIN answer with examples, your Every linear transfor mation T:…

A:

Q: 6. The set of all 2 x 2 matrices of the form " 1 a. is a vector space. b. is not a vector space…

A:

Q: Suppose A is nxn mabrix Cequre mabria) prove that if the pivobe for the matrix is Unique ornoto…

A:

Q: If S represents some orthonormal basis for a four - dimensional inner product space and if (u), =…

A: Let's find.

Q: 1. Consider the set of all 3x3 matrices. Prove that with standard addition and standard…

A:

Q: (a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of inner product…

A: For the first question we will use the key property of isomorphism that the linear transform if the…

Q: Prove that a unitary transform leaves the inner product invariant. That is if U is unitary, then…

A: Please check step 2 for the solution.

Q: Prove that if L is an invertible lower triangular matrix, then L-l is also lower triangular.

A: handwriting solution