Let f be the function from {a, b, c} to {1, 2, 3} such that f (a) = 2, f (b) = 3, and f (c) = 1. Is f invertible, and if it is, what is its inverse?

Q: Let T be the linear operator on R defined by T(x,,x, x) = (2x1, x, - x, 5x, + 4x, + x,). Then 7 is…

A: First express the linear operator in matrix form and then use the matrix to calculate the inverse…

Q: Please solve this question

A: Note: Properties of trace: Let A and B be two matrices. trace(A+B) = trace(A) + trace(B). trace(αA)…

Q: Let ⁄2 be a bounded interval. For any g € L²(2), define the operator T, by T,(f) = f Then T, is a…

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Q: Find the value of x for which A is not invertible: x – 1 A = 1 x + 2

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Q: Let A and B be two sets and f be function. f: A B.Assuming the function f is invertible, what is the…

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Q: Show that the functions f, (x) = 4x" , and f2(x) = x"** are linearly independent on (-00, 00).

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Q: Exercise 0.3.23. (a) Prove that the function f(x) : = vex over [0, ∞). 1+ex is strictly con-

A: As per the policy, we are allowed to answer only one question at a time. So, I am answering the…

Q: Prove the Given theorem:  Let f : X → Y and g : Y → Z be two invertible functions. Then gof is…

A: To show that

Q: Prove that δ: M2×2(F)→F is a 2-linear function if and only if it has the form δ(A) = A11A22a+…

A: Consider a matrix of order 2 as,

Q: 3. Let the functions f: A → B, g:B –→ C, h:C → D be defined in the figure below. A f B g h D…

A: First we will determine which of the functions among f, g and h are onto. Then we will check if any…

Q: Consider the linear operator defined by Show that T is compact. E1 E2 E3 T:l² l², x = (1, 2, 3, ...)…

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Q: Let f(x) = 3x² and. P = {–2, –1, 0, 1, 2, 3}. Compute Lf(P) and U,(P). %3D %3D

A: Consider: fx=3x2 and P=-2,-1,0,1,2,3. Compute LfP and UfP.   If P is a partition such that…

Q: Let f, 9, h be functions from R to R, with f and g invertible. Which of the following is NOT always…

A: Given f,g,h  be functions from R to R, with ƒ and g invertible. We find which of the given above is…

Q: Let L:R R be a linear operator such that L X2 X3- X2 then Ker(L) spanned by: Select one: (e, e )

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Q: Let L:R → R3 be a linear operator such that L( X2 X1 then Im(L) spanned by: X3 X1 Select one: a. O…

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Q: Let A = → Rm be (a;j)ij be an m xn matrix of real numbers, and let f : R" defined by x2 X2 f = A…

A: I am solving this in step 2

Q: Prove that f-(U B: -U(B.), %3D i=1 i=1 and n N (B.). 3=D1 i=1

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Q: Find the value of x for which A is not invertible: x – 1 | 1 A = x2 x +2

A: Consider the given matrix. A=x-11x2    x+2   If A is a invertible matrix then A=0

Q: Let f : A → B and g : B → C be bijections. Prove that gof is invertible and that (g o f)-1 = f-1 o…

A: Problem based on function.

Q: Let f(2) := 25. Show that f is invertible in a neighbourhood of i and calculate (f-1Y(/(i)).

A: Let f(z) = z5, show that f is invertible in a neighbourhood of i and calculate(f-1)' f(i)

Q: Determine the Wronskian of the given set of functions and determine whether they are linearly…

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

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Q: 7. Determine whether (u,v) =3u, v, – 2u, v, is a valid inner-product function or not for R²

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Q: Let L: R2 → R2² be the linear operator defined by L 2

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Q: 1. Let X = {r1, X2, X3, X4} and let b: X x X → [0, ∞) be symmetric with %3D b(x1, T2) = 3, b(x2, T3)…

A: In the given question we have to prove that "h=0 for a function h∈C(X) if and only if h=0." where…

Q: Show that the Jacobian of a function f on E3 is given by the triple product 7.46. af af af J(f) 11…

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Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

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Q: Let L: R→ R be a linear operator such that L X2 )= X3-X2, then Ker(L) spanned by: X3 X3-X1. Select…

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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…

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, ||g||, and d(f, g) for the inner…

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Q: Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S =  [x1, x2] T : x1 ∈…

A: Given a function fx=fx1, x2=x2-x12x2-2x13, where x∈S=x1, x2T: x1∈ℝ, x2<0. (a) Find out the nature…

Q: Consider the functions, f:A→ B and g:B→ A on the sets A-{1,2,3,4,5), B- {3,6,9,12} given by…

A: To solve the following problem of function

Q: Let f1(x) = 3x and f2(x) = ∣x∣. Graph both functions on the interval −2 ≤ x ≤ 2. Show that these…

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Q: Let f(x, y, z) = x2 (e)^y/z. Compute f(-1, 0, 1), f(1, 3, 3), andf(5, -2, 2).

A: Let fx,y,z=x2eyz 1. we have to compute f(-1,0,1) as, fx,y,z=x2eyzHere x=-1, y=0,z=1f-1,0,1=-12e01=1…

Q: Consider the following operator A : L²[0, 1] → L'[0, 1] A(f)(x) = (f(x))² + f(x). If this a linear…

A: Given, the operator  A : L20, 1→L10, 1 defind by                               A(f)(x)=(f(x))2+f(x)…

Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

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Q: 20) Let f, g E F(R,R) be the functions defined by f(t) = et and g(t) = est,r ± S. Prove that f and g…

A: Answer

Q: Show that the functions f1 (x)=e*, f 2 (x)=xe*, and f 3(x)=x*e* are i linearly independent.

A: We will use Wronskian function to check these functions are linearly independent or not. we know…

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: The domain of the function f (x, y, z) = In(6 – 3xy) %3D is {(x, y) E R² : xy 2} {(x, y, 2) E R³ :…

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Q: Find a linear functional ℓ on R3 that satisfies the following conditions: ℓ(1, 2, 2) = 1 ℓ(1, 2, 3)…

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Q: a) Calculus f(x) = x, g(x) = e*, %3D (f, dx b) Let f(x) Example 5, 1 and g(x) = x be functions in…

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Q: Let f(x,y) = x - 5xy -y². Compute f(6,0), f(6, – 2), and f(b,c).

A: Solved below

Q: Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).

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Q: Given that f is an invertible function, if the point (x, y) lies on the graph of y = f(x), then the…

A: We have to find at which point is guaranteed to lies on the graph of  y=f-1(x).

Q: Let f(x) = x and g(x) = ∣x∣.(a) Show that f and g are linearly independent in C[−1, 1].(b) Show that…

A: (a) The given functions are fx=x  and  gx=x. Suppose fx=x  and  gx=x are linearly dependent in C[−1,…

Q: Use the Wronskian to show that the given functions are linearly independent on the given interval I.…

A: For linearly independent, we show Wronskian is non -zero.

Q: Consider the Cobb-Douglas function n f (X1,x2,..,Xn) = x"x. i=1 where x; > 0, and a¡ > 0 for all i=…

A: Consider the Cobb-Douglas function fx1,x2, ... , xn=x1α1x2α2x3α3... xnαn=∏i=1nxiαi Find the…

Q: (b) If f is a bounded linear functional on a Hilbert space H, show that there is a unique y e H such…

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