(a) Let > be a o- Algebra on X. Assume that p1, µ2 :)→ [0, 0) are two measures on X. Is µ1 + kµ, a measure on X? where k is a scalar. Consider the cases: k 20 and k < 0. (b) Let (X, > ,4) be a measure space. For A , B e ). Show that u(AUB) = µ(A) + µ(B) – µ(An B). -

Q: 1. Let e Z(X) and let A = {xe X: f(x)=x}. Show that M is a subspace of X.

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Q: Let X N(u, 1) of size n. Obtain UMVUE for u.

A: Given that  Let X  N(μ,1) of size n. Obtain UMVUE for μ

Q: Suppose Vis finite-dimensional, T = L(V), and U is a subspace of V. Prove that U and U are both…

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Q: Let S = {A2«2| trace(A) = 0}, i.e. S = Show that -a S is a vector space.

A: The solution is given as

Q: (a) Let be a a- Algebra on X. Assume that , lz: - (0, oo) are two measures on X. Is + kuz a measure…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 3. Let W = {(x,y,z): x? + y? = z}. Is W a subspace of V = R3? %3D

A: We need to check if W is a subspace or not.

Q: llnll=max |x C+)| tej show that is norme space. j-[a,b]

A: We have to show that x=maxt∈jxt, where j∈a, b, is a normed space. For this, we have to use the…

Q: If (0,1) ~ (-1,1) and (-1,1) ~ R. Deduce that (0,1) ~R

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Q: 13. Which of the following is a subspace of R4? Justify your answer. a) S = {(a,b, c, d) E R* | a…

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Q: 18)Find curl F and div F if F(x,y,z) = exsiny i + esinz j + e²sinx k

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Q: x + 2y + 2w z + w Let W = x, y, z, w E R be a subspace of Rª. What is the dimension of W? — 3х — 6у…

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Q: (1.7) Example: Let P,(R) the vector space of real functions, define by: = f(x)g(x Is inner product…

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Q: 3 Let v= 3 V = 1 be in R'. Let W = span(v,,v,). Determine if v is in W. 3

A: v = (3, 3,4) , v1= (1, -1, 2) and v3= (2, 1,3)  and  W = span(v1, v2 )  span (v1, v2 ) contains all…

Q: rcise 1.10. (a). Let V = R² = {(x.y)l x, y > 0} with usual addition and scalar multiplication. Is it…

A: Yes, the given set with usual addition and scalar multiplication is a vector space. Let's prove that…

Q: If V(F) is a vector space of polynomials in t of degree at most 3 and D be the differentiation…

A: Given D is differentiation transformation. The basis is 1, t, t2, 1+t3. T(1)=ddt1=0…

Q: u1 O Given a subset U of R² defined by U = |2u1 = u2 Show that U is U2 a subspace of R3

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Q: consider the subspace W of D given by W = span(e^2x, e^-2x) show that the differential operator D…

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Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: Consider the vector-valued function F(t) = exists. 21-6 In(-2) ²³-2t, 2-4t+3/ Find lim F(t), if it…

A: Here we have to find limit of given vector valued function.

Q: Let C[0, 1] be a vector space with inner product defined as (f, Find 9- Olg1=√105 Olg=105 061-…

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Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

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Q: (10) Let the vector space P, has the inner product (p(x), q(x)) = | P(x)q(x)dx (a) Compute ||p(x)||…

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Q: Q1: Recall the definition of 1-1, norm and - norm on R². Consider the matrix A= as a linear map from…

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Q: 5. (10pts) Let A be a positive definite nxn matrix. For any x = R", let ||X||A = √(Ax, x). Show that…

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Q: 10. Let V be the vector space of functions spanned by B = {cos x – sin a, cos a + sin æ}. Find the…

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Q: 1. Suppose V is the vector space of continuous functions f : [-1, 1] → R. (a) Show that (,): V × V →…

A: Given a vector space of continuous functions defined by .(a) To show that given definition is an…

Q: Let S € R² | xy ≥ 0 Y where S uses the usual definitions for vector addition and scalar…

A: Given subset of is the following :Obviously Here noted that vector addition rule : and scalar…

Q: 5. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' = 4f}…

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Q: 3) Let W = { (x, y, z, t) : x = 0, y = -z } be subset of IR*. Is W a subspace of IR* ?

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Q: Using this equation for associated Legendre functions (and for Legendre functions when m=0) 1 d de…

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

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Q: Let A E Mmxn (F). Prove that the column space R(A) is a subspace of F^m.

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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.