b. a Let A = If (x, y) = y" Ax defines an inner product on R? for all æ, y E R² , show that a + c > 2b.

Q: B) Suppose A is an invertible matrix inM, (R). And x, y E R". Prove it: det(A+ ry") = (1+y"…

A: Given: A is an invertible matrix in Mn(ℝ) x,y∈ℝn To prove: det(A+xyT)=(I+yTA-1x)det(A)

Q: For A a square matrix such that ATA = I show that det A = ±1.   If A is square and A5 = 0 show…

A: Given, ATA=IdetATA=detIdetATdetA=1  Since, detI=1detA·detA=1    Since, detAT=AdetA2=1detA=±1 This…

Q: Find the standard matrix of the composition T = T, o Tị for T1 (x, y) = (-x + 2y, x + y, x – y) T2…

A: The given linear transformations T1:ℝ2→ℝ3 and T2:ℝ3→ℝ2 are defined as: T1x,y=-x+2y, x+y,…

Q: Let A and B be m × n matrices that are row equivalent. (a) Prove that rank  A = rank B . (b) Prove…

A: Given :  A and B are m×n matrices with row equivalentNull Space:…

Q: Let U = {u1, u2} and W = {w1, W2} be bases for V, and let P be a matrix whose columns are u1 w and…

A:

Q: Let f1, f2, f3 be functions in F(R). a. For a set of real numbers x1, X2, X3, let (f(x,)) be the…

A: Given that, f1,f2,f3  be the functions in F(R) It is given that fi(xj) be the 3 times 3 matrix.…

Q: Use the definition of an inner product to determine if "⟨u, v⟩ = (Au) ̇(Av) for u, v ∈Rn and A is an…

A: Given: ⟨u, v⟩ = (Au)·̇(Av) for u, v ∈Rn and A is an invertible n ×n matrix. To show if <u, v>…

Q: Find the outer product form of the SVD for the given matrix. (Enter sqrt(n) for vn.) 5 0 A = 0 5 0

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Q: Let (A.x, Ay, As) solve AT I JI Ax Ay I.l - S 0 X As Assume ri + 0, S and X are positive definite…

A: As per the question we are given a linear system. From which we have to find ∆x , ∆y , ∆s And prove…

Q: Let A denote the 2 x 2 matrix We define a function (, ) : R² ×x R² → R by taking (x, y) = y" Ax, for…

A: We have A=1221. We are given the function ·, · : ℝ2×ℝ2→ℝ by taking x, y=yTAx, for each pair of x,…

Q: If A and B are arbitrary real m X n matrices, then the mapping (A, B) = trace(A" B) defines an inner…

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Q: Let (r x s) denote an r x s matrix. Find the sizes of those matrix products that are defined: (c) (1…

A: As per guideline we will only solve the first three subpart of the question number 1 .If you want…

Q: Let A and B be square matrices of order n satisfying Ax = Bx for all x in Rn. (a) Find the rank and…

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Q: Given two skew-Hermitian matrices X and Y , prove XY −Y X is a skew-Hermitian.

A: Solution :- skew hermitian matrix:- When the conjugate transpose of a complex square matrix is equal…

Q: Let a, b, c, d, e, f, g, h be arbitrary numbers. Show that the matrix а 0 0 b 0 0 0 d A = 0 0 e f 0…

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Q: Let C ∈Mm×n(R), and let Y be an n ×p matrix of differentiable functions. Prove (CY )’ = CY’, where…

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Q: Let A e C™×n; that is, A corresponds to a linear operator C" → C*. Define the Hermitian transpose A*…

A: Given: A∈Cn×nA*=Hermitian transpose of A.Ax·y =x·A*y     ∀x,y∈CnTo prove:A*ij =aij

Q: Let f(x,y) =x²+y² +cxy be a function on domain R?; c is an arbitrary constant number. (a) Write down…

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Q: Find the smallest integer value for K so that the linear operator T: R→R such that T(x,y,z) = (3x +…

A: The given linear operator is T : ℝ3→ℝ3 such that Tx, y, z=3x+1+ky, 3y, -ky+z To find: (i) value of…

Q: If A and B are arbitrary real m x n matrices, then the mapping (A, B) = trace(A™B) defines an inner…

A:

Q: Let J, be the n X n matrix whose entries are all equal to 1, and let D(11, An) be the n X n diagonal…

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Q: Use Green's Theorem to evaluate S. axy dx + cxy dy, where C is a counterclockwise oriented…

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Q: 1. Let A = [a;j]4x5, B = [bij]4x5; C = [cij]5x2; D = [d:j]4x2, and E which of the following matrix…

A: Matrix addition: Let A be an m×n matrix and B be an also m×n matrix. A= {aij: 1≤i≤m ; 1≤j≤n} and B=…

Q: Let B = {cos² x, sin x cos x, sin² x}, let V = Span(B), and let L: V V f(x) → f'(x) (a) Compute the…

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Q: Let A be an m x n matrix. Show that: ( a ) Ifxε Ν (Α'Α), then AxE R(Α) Ω Ν (A"). (b) N (AT A) =…

A: Let A be an  m×n matrix. Show that: (a) If x∈NATA, then Ax∈RA∩NAT (b) NATA=NA. (c) A and AT have…

Q: Consider a point xA = (x, y, w)' in image A, and a Fundamental matrix of the form fu f12 fi3 f21 f22…

A: Given: A point in image A, xA=x,y,wT and a Fundamental matrix of the form,…

Q: Let A be a m xn matrix with columns c1, ... , en. Prove that {c1, ..., cn} is independent if and…

A: To prove  c1,c2,...cn is independent if and only if null A=0

Q: Let A and B be n X n orthogonal matrices. (a) Prove that A{AT + BT)B = A + B. (b) Use part (a) to…

A: (a) Since A and B are orthogonal matrices, we have  AAT=ATA=BBT=BTB=I, where I is the n×n identity…

Q: Let f(x, y, z) = x°yz + 3z – 6xy + 2. - a) Compute the Jacobi matrix of f b) Compute the Hesse…

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Q: Consider P (R) with the standard inner product = So f(x)g(x)dx 2 a) If p1 (x) = –8 x – 8 then…

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Q: Let A be p x p matrix such that ATA = A?. Show that: (a) tr (A - AT) (A- A") = 0. (b) Show that A is…

A: Let A be p×p matrix such that ATA=A2a) trA-ATTA-AT=0b) Show that A is symmertric

Q: Let A = CD, where C and D are n × n matrices and C is invertible. Prove that the matrix product DC…

A: A = CD, where C and D are n × n matrices, and C is invertible. We need to Prove that the matrix…

Q: Let Y-Nn(0,E)and let A be any nxn symmetric matrix. Show that m.g.f of Q=Y'AY is M(t) : det…

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Q: Let A € RPXm have rank r. Show that there are matrices C and B such that A = св, where C E RPXT and…

A: Please check the solution in next step

Q: Let f(u) = u™ Au, where u e Rd and A e Rd×d is a symmetric matrix. Calculate Vuf(u).

A: Directional derivative: If the function f is differentiable at x, then the directional derivative…

Q: Let B be an n x n symmetric matrix such that B2 = B. Any such matrix is called a projection matrix…

A:

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

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

Q: Let A be a 2 × 2 matrix, and let LA be the linear operator defined by LA(x) = Ax Show that (a) LA…

A: Both parts can be established by direct calculations for 2x2 case. The same results are true in nxn…

Q: Let A be any invertible n x n matrix. Show that for u, v in Rn, the formula (u, v) = (Au)· (Av) =…

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Q: Let V denote the inner product space of real 2 by 2 matrices endowed with the inner product (A, B) =…

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Q: Suppose A is a 2 x 2 matrix such that for all x, y E R², (Ax) · (Ay) = x ·y Must it be true that A…

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

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: Given two Hermitian matrices X and Y , where XY not equal to Y X, prove XY −Y X is not a Hermitian.

A: Solution

Q: (b) Let A be mxn matrix Such that AA iS invertible Show that the Coloumn ot A are Invertsble.

A:

Q: Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of…

A: Given Ais a symmetric matrix therefore aij=aji…

Q: Let U be an n x n matrix such that UT U=I. Show that U=+-1. (plus or minus 1)

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Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

Q: Find the smallest internal value for k so that the linear operator T: R°→R° given by T(x,y,z) = (x…

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Q: Let A(t) € R"X* be a square matrix whose elements vary in time, but such that |A(t)| # 0 for all t…

A: Let A(t) be an n×n matrix with real entries and depending on the variable t. From the definition of…