1. For the matrix A, let T: R* → R' be defined by T(X) = Af for each i in R'. A is given: [2 1 1 2 A =1 -1 3 3 4. 1 0 -7 -3 -4 6 7 -7 2 a) Find Ker(7) and the nullity

Q: = [aij), whose entries satisfy the conditions fi +j if i sj aij= ( 1 if i >j is: The 4 x 4 matrix 4…

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Q: At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula e Determine e…

A: A detailed solution is given below.

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: 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: Given the bases B=(u, u2} and B'={u'1, u'z} for R2, where U2 and u'2 The transition matrix PR-8 from…

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Q: 5: The Hessian matrix for f(x) = x³x3 + x2x3 at the point [1 2 3] is given by: [6 [18 0 31 0 671 20…

A: Let's solve.

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: Suppose v1, v2, v3 are eigenvectors for T. This means T( vi) = Ai Vi for i = 1, 2, 3. What is the…

A: Given: v1, v2, v3 are eigenvectors for T. This means T( vi) =  λi Vi for i = 1, 2, 3. To find: The…

Q: Given the bases B={u , Uz} and B' = {u'1, u'z} for R, where U2 and u' u'1= u'2 !! The transition…

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Q: A linear operator T: P2 P2 is defined as T(p(x)) = p(2x+1), i.e. T(c + c₁x + c₂x²) = C₁ + c₁ (2x +…

A: Here T is given linear operator from P2 to P2. First we have to find matrix TB of T with respect to…

Q: Let S = {V1, V2} and T = {w1, w2} be ordered bases for P1, where W1 = 1-1, w2 =t+1. 1 2 If the…

A: Given: S=v1,v2 and T=w1,w2 are the ordered bases for P1, where w1=t-1, w2=t+1 Also, the transition…

Q: Let A and B be 6 x 6 matrices, with det A = -10 and det B = 5. Use properties of determinants to…

A: Given that det A=-10 and det B=5 find a) det(3A) b)det(ATB-1)

Q: At by first finding a fundamental matrix X(t) for x' = Ax and then using the formula e At 099 A = 90…

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Q: 8). Compute the Hessian matrix for functions (a) f(x, y, z) = xy³² – ² – y; (b) f(x, y, z) = xyz;…

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Q: 6. Let B be n x n matrix. Define a function T : Mn(R) → M„(IR) by T(X) : XB – BX for all X € M, (R).…

A: A, B and C be n×n matrices, then  A(B + C) = AB + AC (B + C)A = BA + CA

Q: Let S (V1, V2, V3) and T = {W1, W2, W3) be ordere bases for R, where %3D V2 D V3 = 0. Suppose that…

A: please find the answer below

Q: 4. Let f1, f2, fz be functions in F(R). For a set of real numbers x1, x2, X3, let (f(x,)) be the…

A: Note:- Here, you don't mention field TR is ℝ or anything else. So, i'm assuming this ℝ. If it is…

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: Consider the 3 x 3 matrix A and the vector & e R³ given below 0 1 0 1 0 1 0 1 0 1 A = i = Prove that…

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Q: 1. Let x E C" and y E C™, with y # 0. Consider the m x n matrix defined by A = yx*. (a) Show that…

A: Given that x∈ℂn and y∈ℂn, with y≠0. Consider m×n matrix defined by  A=yx*. To prove (a), (b) and…

Q: Let X(t) be a fundamental matrix for x′ = A(t)x on the interval I. Q. If t0 ∈ I, show that the…

A: Let X(t) be a fundamental matrix for x' = A(t) x t0∈I To show: WE have to show that the solution of…

Q: Let T: P3 (R) ⟶ R, given by T(p(x)) = ∫10 p(x)dx. The matrix of T with respect to the canonical…

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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: 5: The Hessian matrix for f(X) = x}xy + x3x3 at the point (1 2 3], is given by: [6 0 31 [3 0 61 [18…

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Q: - x2 Let T : R? → R² be defined by T x2 2 Let B = {u, = ,U2 = -6 and C Find [T, the matrix…

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Q: Generate the matrix A of order n= 100, whose generic element is a; = sin(max(i, j)) The condition…

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Q: 2. Let A E Mn (R) be a skew-symmetric matrix, i.e. AT A. Prove that (a) x" Ax = 0 for all r E R".…

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Q: Given the bases B={u , u2} and B' ={u'1, u'z} for R, where U2 and u' 2 The transition matrix PR-B…

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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 M be the following matrix over Z2: 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 M = Which one of…

A: Given  A 5x4 matrix is given. Nullity = n- rank Nullity = 5 - rank

Q: Let D : Pn(R) → Pn(R) be the differential operator defined by D(f) = f'. Show that the matrix…

A: I have used the fact that non zero nilpotent matrix is not Diagonalisable

Q: Let A be a 3 × 3 matrix such that Atr A = 4I3. Prove that det A = ±8. (Here, I3 is the 3 x 3…

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Q: The matrix has a singular value decomposition A = UEVT where and and The 3,2 entry of Vis (V)32 =…

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Q: Find the Jacobi matrix for the following function. 8x°y- 4y + 3x f(x, у) %3D e^ sin y

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Q: a) Let U be an m xn matrix with orthonormal columns, and x and y be in R". Then I Ux I=I x| (Ux)…

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

Q: Label the following statements as true or false.{a) If E is an elementary matrix, then det(E ) = ±…

A: Since you have posted multiple subparts, as per our policy we will answer first three subparts.…

Q: If o is a normalised fundamental matrix of X = AX at to = 0 and f is continuous in t, then Xp(t) = |…

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Q: 2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV→R given by (X,Y)= tr(XY) (X,Y)=0 for…

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Q: For each of the following matrices A E M,(R), test A for diagonalisabil- ity. If A is…

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Q: 2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X, Y) = tr(XY) is…

A: The complete solutions are given below

Q: 10) Define T E L(R²) by T(x, y) = (41x + 7y,-20x + 74y). Verify that the matrix of T with respect to…

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Q: 1) Let [2 1 1 -7] -1 1 A 3 1 0 0 7] Find the fundamental matrix for y'(t) = Ay(t).

A: Given A=211-70-10100310007

Q: Given the bases B={uj , u2} and B'={u'1, U'z} for R², where U2 and u'1= %3D The transition matrix…

A: We have B=u1,u2 and B'=u'1,u'2 for R2 where u1=10 , u2=01 , u'1=21 , u'2=11

Q: 7. Find the Jacobian matrix of the following. a) x= (u+v),y = (u-v).= = 2uvw b) x=r cos 0, y =r sin…

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Q: Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this…

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Q: 3. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw||? a. If X is N…

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Q: 16. Find the standard matrix of the scaling operator Sk on R", where k e R, given by SkE) = kx.

A: We have…

Q: 1. Let Y = [Y1 Y2]T and X = [X1 X2]T be two random vectors such that Y = AX, where A is a 2 × 2…

A: Let A=a11a12a21a22, it is given that Y=Y1  Y2T=Y1Y2 and X=X1X2T=X1X2 so it follows:…

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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