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: Suppose A is the matrix for T: R3 → R3 relative to the standard basis. Find the diagonal matrix A'…

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Q: Let R⁴ have a product in Euclidean. Use the Gram Schmidt process to convert the basis (u1, u2, u3,…

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Q: Let A € Mnxn (R), & be the standard basis for R", and B = {₁,...,n} be another basis for R". If P is…

A: Given, a matrixA∈Mn×n(ℝ). Given, ε as the standard basis ofℝn and  β={v1,v2,...,vn} as another basis…

Q: Prove the theorem: AAT is a symmetric matrix for any matric A.

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Q: Let {v1, v2, ..., vn} be an orthogonal set in R" with respect to the Euclidean inner product. Let A…

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Q: Suppose A is the matrix for T: R - R relative to the standard basis. Find the diagonal matrix A' for…

A: i just used the definition of matrix of A with respect to basis B'

Q: → Suppose TR³ R³ be a linear bronsformation whose standard matrix has the basis (o 1, 2), (2, 1…

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Q: Denote the column vectors of the matrix 11,922.. , ann +] True, false, or uncertain: a; is…

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Q: Exhibit a basis for T2, the vector space of all 2 x 2 matrices of real numbers with zero trace.

A: Let V be a vector space over a field F. A set S of vectors in V is said to be a basis of V if  (i) S…

Q: 5. Consider the 4 x 5 matrix A = [u₁|u2|u3|u4| u5], where the columns are U₁= 2 , U2 = 5 Uz = -1 -2…

A: The  4×5  matrix  A= u1 | u2 | u3 | u4 | u5   where  u1=1234 , u2=5102 , u3=   2-1-2   3 , u4=  1…

Q: Suppose {u₁, U2,..., U2n} is an orthonormal basis for R2n. Let n A = Im - Σuju?. j=1 Show that (a)…

A: We show that A is idempotent matrix   For this we have A2 =A .. And the eigen values of idempotent…

Q: Taylor is paid on the frst day of the month and $80 is  automatically deducted from her pay and…

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Q: 2. Let B = {V₁,..., Vn} be a subset of Fn containing n distinct vectors and let A = [v₁... Vn] be…

A: Let B = {v1,..., vn} be a subset of Fn containing n distinct vectors and let A = [v1, v2,... vn] be…

Q: Prove that the matrices and a a a 0. a 3. a are Similar ( These 2 matrices are not deagonalısable)

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Q: Let A and B benxn matrices such that AB = 1. Prove that |A| +0 and |B| + 0.

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Q: 4) Let M₂,2 denote the vector space of 2x2 matrices and W be the set of all 2x2 matrices A= such…

A: Recall  the  definition  of  a  basis  of  a  vector  space :  Let  V be  a  vector  space  over a…

Q: V has bases B = (v1, v2, v3) and B' = (w1,w2, W3). If wi vị + 2v2 + 4v3, wz = 2v1 + 5v2 + 6v3, wz =…

A: Given that, B=(v1,v2,v3),B'=(w1,w2,w3) Given that,…