4. Let the vector space P([-1,1]) have the inner product = f(t)g(t)dt, using 21

Q: Show that λis a root of char(x) if and only if there exosts a non-zero vector v in V with L(v)=λv

A: Assume that λ is a root of charL(x) then | λIdv–L | = 0. It is known that, the system of Bx = 0 has…

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

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Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given  be the set of all pairs  of real numbers together with the following operations:

Q: Prove that V is a vector space with the operations

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Q: Let P₂ be the vector space of polynomials of degree at most 2. Let T: P₂ → P₂ by T (p(x)) = p(x) + x…

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Q: 1. (18 points) Let u(t) be a differentiable vector function such that |u(t)| = 1 for all t. You may…

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Q: Consider the vector-valued function [f(x, y) xt – xy² [g(x, y) 2 Vx + y³x² [f(x, y) Which of the…

A: Find the Jacobian of the given matrix

Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies  T:U-V  Then T(u+v) = T(u) +T(v)          T(au) = aT(u)  For all a…

Q: 2s Let H be the set of all vectors of the form Find a vector v in R' such that H span {i}. -s S

A: Spanning set of H

Q: Find the domain of the vector functions, r(t), listed below. You may use "-INF" for -∞o and use…

A: According to bartleby guidelines we supposed to do the first one of multiple questions kindly repost…

Q: 21 Determine the domains of the vector-valued function r2(1) = (v/8 – 13, In 1, eviy %3D -

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Q: Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients,…

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Q: Find the domain of the vector function T (ty= Vt + 7 Domain: {t <t< }

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V × V → V…

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Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

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Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: xy -) zi – xk -) xỉ + zk

A: