Suppose that U is a 3-by-3 unitary matrix. Prove that U is a normal matrix and prove that there exists an orthonormal basis u1, U2, U3 of C° and complex numbers C1, C2, C3 with U = c¡u¡u* + c2u2u" + c3U3u such that |c|| = |c2| = |c3| = 1.

Q: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

Q: Prove that if R is a matrix in echelon form, then a basis for row(R) consists of the nonzero rows of…

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Q: i) Find the matrix of T with respect to ordered basis B' = {(1, 1), (1, – 1);:.

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Q: Let A be an n×n matrix. Prove that A and AT have the same Jordan canonical form.

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Q: In the vector space P, of polynomials of degree atmost two, the first column of the matrix of…

A: The first column of the transition matrix obtained from the expression of first vector a1 with…

Q: Find a unitary matrix U such that UH AU is diagonal, where (: 4 A = -3 2i 0 -2i

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Q: Consider the matrix mapping A:R5-R³ where [1 0 0 0 2 A = |0 2 2 1 0 Lo 0 2 0 Find a basis and the…

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Q: In the vector space p of polynomials of degree atmost two, the first column of the matrix of…

A: We have to choose the correct answer from the given options according to the given information

Q: In the vector space P, of polynomials of degree at most two, the matrix of transition from the basis…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Given bases B = {(0,1, 1), (1,0, 1), (1,0, – 1)} = {u1, U2, U3} and B' = {(0,0, – 1), (0, 3, 1),…

A: The change of basis matrix is given as follows :

Q: Find a basis of the linear space V of all 3 x 3 matrices A for which both | - A 1 and are…

A: We have to find the basis of linear space V.

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: Suppose A is a complex matrix. Show that AA and A" A are Hermitian.

A: Explanation of the answer is as follows

Q: Prove that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

A: Given that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

Q: Let U be an n xn orthogonal matrix. Show that if is an orthonormal basis for Rn, then so is…

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Q: Let S be the standard basis for IR and consider the non-standard ordered bases of R4 1 3 -1 3 1 -2…

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Q: Let V be R³ and TB = {n₁, n2, n3} and SA = {m₁, m2, m3} be the ordered basis fo 3 2 2 n₁ = - A , n₂…

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Q: Let A be an m × n matrix with rank m. Prove that there exists a n × m matrix B such that AB = Im:

A: Take any n x m matrix B with rank n. We know that rank AB = rank A = m (for n>m) hence AB is…

Q: Prove that the n X n identity matrix In is invertible and that

A: We have to prove that the n X n identity matrix In is invertible and that In-1=In.

Q: Let T : R? → R´be defined by [2 _5||X1 X2 1 6] [x2] а If B = is a basis for R“ such that |T'|B is a…

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Q: If A is a n x n matrix with NON zero determinant, then its columns are a basis for a n dimensional…

A: “Since you have asked multiple questions, we will solve the first question for you. If youwant any…

Q: In the vector space p, of polynomials of degree at most two, the matrix of transition from the basis…

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Q: Suppose A E M,xn(C) is a normal, invertible matrix. Prove that A A- is unitary.

A: Given A € Mnxn(C) is a normal matrix.

Q: Let M be a matrix of integer coefficients. Prove that M-1 exists and has integer coefficients if and…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find a basis of Null(A), C(A), and C(AT) for each of the following matrices [1 2] 3 4 (i) A 1 2]…

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Q: . Prove that (i) . =1, and (ii) ||, =\n (I is the Identity Matrix of order n).

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Q: Let A be an m ×n matrix with rank m. Prove that there exists an n ×m matrix B such that AB = Im.

A: Let A be a mxn matrix with rank m. To prove that there exist a nxm matrix B such that AB = Im.

Q: Let Q be an n × n orthogonal matrix. Show that cond2(Q) = 1

A: Q be an n × n orthogonal matrix. To show: ||Q||2=1

Q: Let A and B be two square matrices of order b x n. Prove that A and B are non singular if and only…

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Q: prove that if A is an invertible matrix then AT is also invertible and (AT)-1= (A-1)T

A: Assume a matrix B. Such that B=A-1

Q: Find the matrix A, relative to the standard basis {e1, e2, e3, e4}, of t

A: Given, W be the subspace of ℝ4 spanned by the vectors                                    1-11-1 and…

Q: Let U be an orthogonal matrix. Prove that det(U) is either 1 or –1.

A: Orthogonal Matrix: A matrix A is said to be an orthogonal matrix if AAT=I where I  is the identity…

Q: Find the coordinate matrix of x relative to the orthonormal basis B in R". {G ) (-) 0, 0, 1)} x =…

A: Given vector is x=20, 10, 25 and the corresponding orthogonal basis is B=35, 45, 0, -45, 35, 0, (0,…

Q: Prove that if A is an invertible matrix then det(A−1) = 1/det(A)

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Q: Prove that the n X n identity matrix In is invertible and that ln-l = In

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Q: Prove that if A is a positive definite matrix and A = UΣV* is a singular value decomposition of A,…

A: A  is a positive definite matrix and

Q: Let B = uj = ,U2 = and C = , V2 = Find Pc, the transition matrix %3D from the standard basis of R2…

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Q: Let B = {v1, v2, V3} be a basis for R3. For any u, v E R³ define || (u, v) = [u]E[v]B• (a) Is (, )…

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Q: Find the coordinate matrix of x in R'" relative to the standard basis. = (2, –6, –5, 9) [x]s =

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Q: 3. Prove that (i) /, =1, and (ii) ||, =Vn (I is the Identity Matrix of order n).

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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: (4) Let A be an invertible matrix. Prove that if A = Q¸R1 decompositions of A and if the diagonal…

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Q: Find a basis B of P3, the vector space of polynomials of degree < 3, so that the transition matrix…

A: The standard basis of the vector space of polynomials, ℙ3 of degree ≤3 is,1,x,x2,x3 And the…

Q: In the vector space P, of polynomials of degree at most two, the third column of the matrix of…

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Q: 6. Prove that the Rotation Matrix R corresponding to quaternion q = (a, b, c, d) is given by 1. R=…

A: Hint:    Consider a 3×3  matrix      a00a01a02a10a11a12a20a21a22

Q: Let A be an n x n matrix. If vi, v2,..., v are vectors in R" such that {Avi, Au2,..., Avn} is…

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Q: Given that B = { [1 -2 3], [0 1 0), [0 2 1]} is a basis of R and C = {[-3 -5 6], [2 3 -3], {[- 3 [0…

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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: For the matrix A = 120 1 1 3 UH AU is upper triangular find a unitary matrix U such that

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Q: Assume that T and S are matrix of the same size. Prove or Disprove that (T + S) 2 is a symmetric,…

A: Given   T  and   S are matrix of same size. We have to prove if  T+S is symmetric, skew - symmetric…