Ques. 3: Check whether the operator T: R2 → R² defined by T(x,y) = (y,x) is linear. Moreover if T is linear find T if exists.

Q: 4) For which one of these linear maps T:R → R? is det T > 0? a) T(1, y) = (y, -r) b) T(1, y) = (x,…

A:

Q: Consider the following linear operator; T : R → R° given by T(x, y, z) = (y, x + z, z).

A: We have to find the real Eigen value of T

Q: . Show that the operators T₁,, T4 from R2 into R2 defined by (1, 2) (1, 0) (§1, §2) (0, §₂) (§1, §2)…

A:

Q: 4. For the linear canonical system x' = Jx, x € R², determine the nature of the origin as a fixed…

A: Given canonical system x'= Jx  Case-1:-  det >0 if both Eigen value positive:- The product of the…

Q: Suppose , t2, ., , are linearly independent. Prove that S is linearly independent where (a) S=…

A:

Q: Define: T:P, → R by T(P) = [ p(x)dx. What is the kernel of T ?

A: Given that: T:P2→RTp=∫01px dx

Q: If |r(t)| = c (a constant), then r' (t) is orthogonal to r(t) for all t.

A:

Q: Let x = s e and y =t es. Compute the Jacobian

A: Given: x=set and y=tes.

Q: Let T:R2 R2 be a linear transfomation define as T T()=(;) and T(,) = G) Hind TC)? Find?

A:

Q: Define an operator T: fºº → f° by T(5₁) = (¹) # and bounded. for each (₁) E. Show that T is linear…

A:

Q: The linearization at a = 0 to V6+ 5x is A+ Bx. Compute A and B %3D A = %3D B = %3D

A: Consider the given: Let fx=6+5x    at x=af0=6+50f0=6+0f0=6

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, 32, ..., Yn)(to) #0…

A: If two functions are linearly dependent then the Wronskian is zero for every value of t.

Q: Consider the linear system x'(t) = 1 x(t) defined for t > 0. Verify that the following (G +) 1+]…

A:

Q: Find the EXACT Jacobian a(x,y) d(u, v) 1 1 - -— (9u - 4v), y = —— (3u +3v) 4 4 X = for the following…

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Q: [2 Lo 2. and defined T: R² → R³,by T(X) = AX. Let A Find the images under T of u = |E|, and v =

A: Given a linear transformation T:R2→R3 by T(X)=AX where A= 2002

Q: Are the functions f(x) = e ^x and g(x) = e ^|x| linearly independent on I = (−1, 1)? Are they…

A:

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = x2 + y2 + z2…

A: given that a function f(x,y,z)=x2+y2+z2 we need to find the linearizations of f(x,y,z) at (1,1,1).…

Q: Let T: R² R2 be a linear operator defined by Find a formula for T I Y T[H]-[2] + [2] = [3]. 5

A: To find the formula for the linear transformation of [x y].

Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: Consider x1 = et and x2 = v(t)et. If x1 and x2 is linearly independent, then v′(t) = 0. Is this…

A:

Q: Are the following linear maps invertible? a) T: R3 → R³ defined by T(x, y, z) = (y, z, x)?

A:

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A!, 2.11. Ax(1) = x(1)…

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

Q: (b) Show that the Linear guen Operators. are not with T(f)= Lf6), f()+1].

A:

Q: The linear operator T: R2 --> R2  defined by T (x; y) = (4x + 5y; 2x + y) is it diagonalizable?

A:

Q: X = R² and = = { $₁x} V { \\x>0} where U4 = {(x,y) | x² + y² <²} Prove (x, t) is Topology

A:

Q: Say the linear operator given T:R³ → R³, defined by T(x, y, z) = (x + y, x – y + 2z, 2x + y – 2). I…

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Q: If T: H→→H is a bounded self-adjoint linear operator and T‡0, then T"#0. Prove this (a) for n = 2,…

A: Given T:H→H is a bounded self-adjoint linear operator and T≠0. We know that if T is a self adjoint…

Q: Consider the bounded linear operator T:C0,1]C0, 1] defined by -Ta =y y(t) = ta(t). Enter the norm of…

A: Given that T:C0,1→C0,1 is a bounded linear operator. Defined by x↦Tx where x is continuous function…

Q: 1 if i= j 0 otherwise. P:(c;) = 5 is a linearly independent set.

A:

Q: Show that the functions y1= e^t,  y2 = e^-t, and te^-t are linearly independent.

A:

Q: Show that the operators T,··, T4 from R² into R² defined by ... (§i, 52)→ (Š1,0) (§1, É2)→ (0, E2)…

A:

Q: s the linear operator T : R3 --> R3 defined by T (x, y, z) = (2x - 3y + 7z, 5y + z, -z)…

A:

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = x2 + y2 + z2…

A: To find the linearization Lx,y,z of the function at the given points fx,y,z=x2+y2+z2 at 1,0,0

Q: Define T:R → R' by T(x, y, z) = (x-5y+4z, y- 6z). a) Is T one-to-one? Justify your answer. b) Is T…

A:

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: Question 5. Let T : V → V be a linear operator satisfying T² = T. Define U1 = {v € V : T(v) U2 = {v…

A: Given: U1=v∈V:Tv=v U2=v∈V:Tv=0 T2=T Objective: To show that V=U1⊕U2 That means to show that every…

Q: Let T be function from R to R2 de fined as T (x) = (0, x), Then show that if T is linear operator or…

A:

Q: 8. Show that an isometric linear operator T: H→→ H satisfies T*T= I, where I is the identity…

A: Let T : H → H  be an isometric linear operator. We need to show that , T satisfies T* T = I We know…

Q: suppose that x,y,z∈R3 are such that x.(y×z)≠0. prove that x,y,z are linearly independent.

A:

Q: The functions y1, Y2, · . . , Yn are linearly independent, if and only if W (y1, 22, *. Yn) (to) # 0…

A: Consider the following differential equation, y''+pxy'+qxy=0

Q: Let L: R → R be a linear operator such that L X2 )=| X1 then Ker(L) spanned by: Select one: а. b.…

A: Given a linear operator L from R3 to R3. We need to find the set of vectors that spans Ker(L)

Q: 2: Let a function d: R2 x R2 R defined by d(x, y) = ((x1 – y1)² + (x2 – y2)²)* %3D = XA Vx =…

A:

Q: If W(f, g) = sin t then the functions f, g are linearly independent Select one: O True False

A:

Q: Show that the functions f (x) = |x + 1|and g(x) = x² are linearly %3D independent on the interval I…

A: Given that f(x)=|x+1| So, f(x)=x+1 in [1, ∞) g(x)=x2

Q: Ques. 3: Check whether the operator T: R2 → R2 defined by T(x, y) = (y,x) is linear. Moreover if T…

A: A mapping T:R2→R2 is a linear transformation if and only if a) T(u)+T(v) = T(u+v) b) T(cu) = cT(u)…

Q: Show that each linear operator T on R2 is nonsingular and find a formula for T- , where (a) T(x,y) =…

A: We have to show that linear operator T on R2 is nonsingular andfind the formula for…

Q: Let L:R→ R be a linear operator such that L X2 )=x1, then Im(L) spanned by: -> X3 Select one: a. 一 O…

A: The given linear operator is: L:ℝ3→ℝ3 such that Lx1x2x3=x1x1x1

Q: Let ƒ (x, y) = x² - y² . Find all all points (x, y) such that Vf(x,y) =±(1, 1).

A:

Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…