. Let B = [X³, X², X, 1] and C = [1, X, X²], which are ordered bases for P3(R) and P2 (R), respectively. (Note the orders of these bases!) Define T: P3(R) → P₂(R) by T(f) = f'. (a) Compute c[T] B. (b) Let f(X) = 4-5X²+2X³. Compute B[f]. Compute T(f). (c) Define S: P₂ (R) → R² by S(f) = (f(-1), f(1)). Compute D[S]c, where D is the standard ordered basis of R². Compute D[ST]B, the matrix of the composition ST with respect to the bases B and D.

Q: Prove the shifting theorem F[s(t + to)](w) = exp(+iw to) F[s(t)](w).

A: The Fourier Transformation of a function is given by;,   where The objective is to prove the…

Q: 10. Define a function T: P(R) → P(R) by T(ƒ) = f' Prove that T is linear and onto, but not…

A:

Q: 6. Determine whether the following set of polynomials is linearly independent or not. (1 – t)?, t –…

A: Solution:-

Q: 6: F7 → F2 where (x1, x2,...,x7) = (X1 + X2, X3 + x7). Prove is not a linear functional on F7

A:

Q: (a) Prove the set (A(*)./a(*),fa(a)) is linearly independent. (b) Use Gram-Schmit process to find an…

A: Given,     f1(x)=3,  f2(x)=1-x, and f3(x)=x2+2x-1

Q: et P E L(C") be defined by, P(z1,..., zn) = (zn, Z1,.….., Zn-1) Determine whether or not P is…

A: Let P∈L(Cn) be defined by   P(z1 , . . . , zn) = (zn , z1, . . . , zn-1)Determine whether P is…

Q: Compute [ f(x,y) dA, where f(x,y) = y²√ and D is the set of (x, y) such that x>0, y> x², and y<10-…

A:

Q: Let B = {1, t − 8, (t − 8)²} : - and C = {1, t − 5, (t − 5)²} - be two bases for P2. The function…

A:

Q: Let B = {1, t − 3, (t − 3)² } and C = - - {1, t − 2, (t − 2)²} - be two bases for P2. The function…

A: a=1b=-2c=5Explanation:

Q: I need b answered

A:

Q: Let T: M22 → P₂ be defined by T T([1]) ¹([8]) T([1₂8]) T([i]) −3x² + (−1)x+ (1). :52+(−1)2+(1) MDB…

A:

Q: Let T: R² → R² be defined by T x1 = x1 x2 Let u = [2] and C = V1 = , V₂ = [A]} x1 Find T(u), the…

A:

Q: Let G : R? → R³ and F : R³ → Rª be defined by G( (x, y)) F( (a,b, c) ) (x +y, x – y, 2y) (a + b+ c,…

A: Given G:R2→R3 is defined as Gx,y=x+y,x-y,2y Also,  F:R3→R4 defined by Fa,b,c=a+b+c,2a+c,b+2c,a+b.

Q: Define T:P,-R as shown to the right. P(- 1) a. Find the image under T of p(t) = - 2-t b. Show that…

A:

Q: fi(x) 1 A = {fi, f2, f3} c F(R) and f(x) = f:(x) = |x+ 1| PROVE: A is linearly independent. (A L.I.)

A:

Q: Suppose S, T € L(V) are such that ST = TS. Prove that N(S) and R(S) are invariant under T.

A: It is given that S,T∈ℒV are such that ST=TS. It can be observed that for any v∈NS, Sv=0. Since,…

Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

A:

Q: Consider the basis S - (V₁, V₂} for R², where v₁ - (1, 1) and v₂ - (1, 0), and let T:R² R² be the…

A:

Q: Let f : R3 → R³ and č: R → R³. Assume f(x, y, z) is such that -2 D{(1,0, 2) 3 -1 -2, If c(t) = (t²,…

A:

Q: Define briefly u(t) and 8(t)

A: u(t): u(t) represents the unit step function, which is defined as u(t) = 0 for t &lt; 0 and u(t) = 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: Let L:R→ R be a linear operator such that L X2 )= X1 then Ker(L) spanned by: X3 X1 Select one: a.…

A:

Q: 2. Let T: P2(R) → R³ and 4 2 [T]; 1 1 1 where B = {1, x, x²} and y = Is T invertible? {(1,0,0), (0,…

A: The solution is given as

Q: Let B = {1, t − 3, (t − 3)² } and C = - - {1, t − 2, (t − 2)²} - be two bases for P2. The function…

A: a=1, b=-2, c=5Explanation:Step 1: Step 2: Step 3: Step 4:

Q: Suppose that f: X → Y and let AC X and BCY. a counterexample: counterexample: a (a) Prove or give…

A: (a) Let, X = Y = {1, 2, 3}              A = {1} , B = {2, 3}      Let, f(1) = 2 = f(2), f(3) = 3.…

Q: (a) Let f, 9, h : N → R²º. Prove or disprove that if f + g€ O(h), then f e O(h) and gE O(h). (b) For…

A:

Q: Compute [GL2 (R): SL₂(R)].

A: To find: The value of GL2ℝ : SL2ℝ .

Q: The function p(t) = 5 + 7(t − 9) + 2(t − 9)², can also be written as p(t) = a + b(t− 3) + c(t − 3)²…

A: We have to find the value of a,b and c using the given data.

Q: The linear operator T : R³ → R³ is defined by T(x₁, x2, 3) = (W₁, W2, W3), where w₁ = 2x₁ + 4x2 +…

A:

Q: Define T by T: Pn → R by T(p) = =S² p(x) dx, where P is the set of all polynomials whose degree less…

A:

Q: If A%3 2x3 -Зх +2:х € [О, 2) then in (R, T,), A %3D а. [-3, 1] b. [-1, 1] с. [1, 3] С. d. [0, 4] e.…

A: Open sets and Closed sets: The sets in topology τ are called the open sets of the topological space…

Q: Let T : R3 → R* defined by T(v,. Vy, 2,vg) (3, Vz, v1). Find T’(v) and T10° (v)

A:

Q: I need c answered

A: