1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x3 - 2x + 3, a = 2

A: We find f(x) and f'(x) at x=2

Q: (b) Let f(t) = (t- 2)*(t +1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: III) Suppose that the linear map T: R-R' satisfies 00 0-0-0-0-0-0 2 2. What are det T and trace T?

A: Given :-  Suppose that the linear map T : ℝ4 → ℝ4 satisfies T 1000 = 2622 ,  T 0100 = 1311 ,  T 0010…

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: Let T: P4(R) → P4(R) be defined by T(f(x)) = xf"(x). Find a Jordan canonical form of T.

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Q: 6. Compute the norms for (a)||f()||, (b) ||F(x)||2, (c)||f(x)||, where f(x) = -V1-r", defined on [0,…

A: Given

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

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Q: 1. Prove that there does not exist a linear map from Cô to C² whose null space equals {(x1, x2, x3,…

A: Given: Null space = x1, x2,  x3, x4, x5, x6 ∈ ℂ6 : x1 - x2 = 0, x3 = -x4 = x5.

Q: f) Every continuous map T² RP2 is homotopic to the constant map.

A:

Q: Let m, n e Z4 and X + 0. a. If m < n, define an injective map f : Xm → x". b. define a bijective map…

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Q: Prove that the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

Q: Prove that the map f : Zn → Zm defined by f([x]n) = [x]m is a well-defined function if and only if m…

A: To Prove- Prove that the map f : Zn → Zm defined by fxn = xm is a well-defined function if and only…

Q: 2) Use the Wronskian to show the three functions are linearly independent. f(x) = x, g(x) = cosx,…

A:

Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Let T'=Tt:L20,1→L20,1 be an operator for a fixed t≥0. To prove that this operator is a strongly…

Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C° (R). x In x are linearly independent in…

A: Concept:

Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

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Q: 5) Define T: R? - R² by T((x,y)) = (0,y) for all (x, y) E R?. Prove is linear transformation of R?…

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Q: Let T: P3 - P, be the linear map defined by T(p(x)) = xp'(x)-2p(x) (here p'(x) is the derivative of…

A: To solve the following problem

Q: 3) Prove that the multiplication map · : Z/nx Z/n → Z /n given by [x] · [y] = [xy] is well-defined.

A: To prove that the multiplication map · : ℤn×ℤn→ℤn given by    x·y=xy  is well-defined.

Q: Let Pn be the set of polynomials of degree at most n. Given the linear transformation T: P2 → P3…

A: Here, we are given the transformation to be T(p(x))=xp(x). And now, notice the options. Observe that…

Q: Let T : R5 → Rª be a function defined by x1+ 2x2 – x3 - X2 X4 + x5 T X3 X1 – x3 + 5x5 2.x1 + 3.x3 -…

A:

Q: 8. Prove the following properties of the Fourier convolution: (a) f (x) * g (x) = g (x) * f (x), (b)…

A: As per our company guidelines we are supposed to answer only first 3 subparts, kindly repost other…

Q: (b) Consider X = C[0, 1] with the norm %3D = max |x(1)), re[0,1] and let T: X X be the mapping given…

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Q: For a continuous transformation ƒ : X → X, prove that f is topological- transitive if and only if…

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Q: Let AC C([-1,1]) be defined by A = {f€c'(I-1, 1), f(0) = 0, f'(x)| < 1, V x€ (-1,1)} Prove that A is…

A: The set A is defined as A=f∈C1-1,1, f0=0, f'x≤1, ∀x∈-1,1. To prove that A is relatively compact.…

Q: 1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is…

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Q: Prove that the family of operators T (t): L2 (0,1) → L2 (0,1), t > 0, where T (t) p (x) = 0 for 0 <r…

A: Fix an operator Tt:L20,1→L20,1, for a fixed t. To prove that Tt is a strongly continuous semigroup.…

Q: dp(z) Let P, be the set of polynomials of degree at most n and p'(x) = dz L: Pn → Pn 1 defined by…

A:

Q: Let f : (0, 1] → (0,1) be the function defined by f ) = „ for all n E N, and f(x) = x if æ E (0, 1]…

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Q: Let Qc(x) = x2 + c. Prove that if c 1 such that Qc is topologically conjugate to Fµ(x) = µx(1 − x)…

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Q: Let R-R3 be a linear map defined by L(x.y.z) = (x + 2y, 2x+ 3y+ z,x+y+z)- Then what is the kernel…

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Q: Question. 4 Determine the Kernel, nullity and rank of the linear mapping T:R³ R³ as follow. x T(=) =…

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Q: Obtain the cdf of X. Flx) = {x° (10 – 9x) Osxs1 x > 1 Graph the cdf of X. F(X) F(X) 1.0 1.0 0.8 0.8…

A: Introduction:  Cumulative Distribution Function:  CDF of continuous random variable is defined as…

Q: The elementary symmetric polynomials in three variables are given by: S1(х, у, z) %3D х + у+ z,…

A: Firstly, we note that  is the expansion of  in terms of the symmetric polynomials This is a well…

Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

A:

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = x + 1/x, a = 1

A: Given fx=x+1x

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

A:

Q: 6. Let f : R2 → R defined by f((x1 , x2 )) = x1 - x2 . Then determine if is (a) Lipschitz…

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Q: 4. Find the kernel and image of the linear map f(x,y,z) = (2x - z, y + 5z). Also, find the nullity…

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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: Let F: IR3 →IR3 is a diffeomorphism and M is a surface in IR3 , prove that the image F(M) is also a…

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Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: Find the Wronskian for the set of functions.{1, x, x2, x3}

A: we have to find the wronskian for the set of functions. {1, x, x2, x3}

Q: 4. Show that the function f : X → Y defined by f(X) = X U {a} vX €x is a bijection.

A:

Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER:

Q: §3.3, Exercise 16. Let P: R" → R™ and Q : R" → R™ be linear mappings. Prove that S: R" → R" defined…

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Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2

Q: Q4: Let G = (f. f2. fa. fa) with f = 1,2,3,4 are mapping on R- (0} 3 f(x) = x,f(x)= -x %3D %3D A(x)…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.