2 -1 i: Consider V R³, and Let A -1 2 -1 Define (u, v) || -1 2 uT Av. Show that S equipped with this operation is an inner product space Let x e R and decide whether the following matrix is orthog- sin x ii: COS x onal: -sin x COS x

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x°, -1+a…

A: given B=1+x-2x2 , -1+x+x2 , 1-x+x2B'=1-3x+x2, 1-3x-2x2 , 1-2x+3x2 b) find the coordination matrices…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x*, –1+x…

A: The solution is given as follows :

Q: Let B= {cos² x, sin r cos x, sin? x}, let V = Span(B), and let L: V f(x) → f'(x) V (a) Compute the…

A: Here we use the linear combination of basis functions

Q: [a b] (e) Let B be any 2 x 2 matrix. d [cos a] sin a = U11 (i) Show that there are real numbers u11…

A:

Q: (a) Let Pn(R) denote the vector space of polynomials of degree at most n, where addition is the…

A: Given :  T : P3(ℝ) → P2(ℝ)  defined by                  T(p(x)) = 3p''(x)-2p'(x) To find :(a) We…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x = 2a°, –1+…

A: Hello. Since your question has multiple parts, we will solve the first three parts for you. If you…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2a², –1 + x +…

A: Ans a) Consider B={1+x-2x2,-1+x+x2,1-x+x2}B'={1-3x+x2,1-3x-2x2,1-2x+3x2} where P(R) be the vector…

Q: a (e) Let B = be any 2 x 2 matrix. COs a = U11 (i) Show that there are real numbers u1 and a such…

A: We will use the basic knowledge of trigonometry and matrices to answer both the questions you need.…

Q: [a b (e) Let B = be any 2 x 2 matrix. [cos a = u11 [sin a (i) Show that there are real numbers un…

A: This is a multipart question having two or more interrelated subparts (here we have four such…

Q: Consider the following matrices over R, where 0 is a real number: cos 0 - sin 0 cos 0 cos 0 sin 0 Ro…

A: According to question given thatRθ=[cosθ-sinθsinθcosθ], Tθ=[cosθsinθsinθ-cosθ]

Q: 10. Let A be an n x n matrix such that for every x ER", ||Ax|| = |x|| (the norm here is the one…

A:

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2a?, –1+r +…

A: Question 2(b) We need to find the coordinate matrices of px=9x2+4x-2 relative to the bases B and B'…

Q: Let A denote the 2 x 2 matrix We define a function (, ) : R² ×x R² → R by taking (x, y) = y" Ax, for…

A: We have A=1221. We are given the function ·, · : ℝ2×ℝ2→ℝ by taking x, y=yTAx, for each pair of x,…

Q: [1 i+l If A = | 2i 3 -i -i+2 then prove that A+ A´ Hermitian matrix -i

A: Given :  We need to show that is Hermitian matrix

Q: et A e C™×n; that is, A corresponds to a linear operator C" → C". efine the Hermitian transpose A*…

A: Part(i): Let, A∈ℂn×n, let Aℂ be the following property, Ax·y=x·A*y ∀x,y∈ℂn Then,  A*ij=aji Now,…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x - 2x², –1+x…

A: As per the company policy i can answer only first three subparts for you,more than that leads me…

Q: Let V be the space spanned byf1 = sin x and f 2 = cos x. (a) Show that gi = 2sin x + cos x and g2 =…

A: We know that Span is a set, the linear combination of elements of which form the whole space

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+r= 2r°, –1+x +2²,…

A: As per our guideline we are supposed to answer only three subpart of a multipart questions. Please…

Q: 10. Let A be an n x n matrix such that for every x E R", ||Ax|| = ||x|| (the norm here is the one…

A:

Q: Let A = a. By diagonalization of A, compute e^. b. Find ao, a1 so that e4 = aoI+ a1A. c. The…

A: Given that A=5-131 Now we have to find eA

Q: matrices over K. For each j, 1 <j<n, the function E defined by

A:

Q: E = {u: [0, 7] - RNu is absolutely continuous. uz (0) = u(7). ù e L²([0. 7]. RN)) with the inner…

A: Given: u,vE=∫0Tu˙t,v˙t+ut,vtdt for all u,v∈E where ·,· denotes the inner product in ℝN To find:…

Q: Let B = {cos² x, sin x cos x, sin² x}, let V = Span(B), and let L: V V f(x) → f'(x) (a) Compute the…

A:

Q: Let V, W be two matrices of the same order. If all entries of V are nonzero, then we say that X is…

A:

Q: 1 ²+1 (a) Find the inner product (f. 9) = (b) Verify the Cauchy-Schwarz Inequality. 2. Let f(x)= x…

A:

Q: Let L be the linear operator on R? defined by L(x1, x2) = (2x1 – 3x2, x1 + 4x2)*:\\ i). Find the…

A: We have to find the matrices A and B which are with respect to the standard basis and 1,1,0,1…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2a², –1+…

A: As per bartleby guidelines for more than three subparts in a question only first three should be…

Q: Let V be the space spanned by f 1 = sin x and f2 = cos x. (a) Show that gi = 2sin x + cos x and g2 =…

A: f1= sin x; f2 = cos x spanned Vg1 = 2 sin x + cos x; g2 = 3 cos xh = 2 sin x - 5 cos x Here we use…

Q: (e) Let B be 2 x 2 matrix. any a CO a = Uj1 (i) Show that there are real numbers u11 and a such that…

A:

Q: (d) Consider the vector space C|0, 1] of all continuously differentiable functions defined on closed…

A: We have solve the integration

Q: Let f: C → C be differentiable at zo. Considering ƒ as a map R? → R², prove that f'(zo) = kR where k…

A: (a)

Q: Define the spectral radius of a bounded linear operator A e BL (X). Find the spectral radius of A in…

A: (b) The largest absolute eigen value of a square matrix or linear operator is the spectral radius.…

Q: Let L be the linear operator on R? defined by L(x1, x2) = (2x1 – 3x2, 21 + 4x2)*:\\ i). Find the…

A: We have to find three matrix A B and P as described in the question.

Q: 8. Let L: R5 → R' be the linear transfor mation defined by 1 0 1 0 2 0 -1 0 0 -1 -1 3 -1 2 -1 -1 1…

A:

Q: Let E be any bounded closed set in the complex plane containing an infinite number of points, and…

A:

Q: 1. Show that every scalar product on R² is in the form: = ar + b(r12 +.r2) + cr22. (0.1) where a.b.c…

A: The main objective is to prove that the scalar product is strictly positive definite.

Q: 2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X, Y) = tr(XY) is…

A: The complete solutions are given below

Q: If A is n x n invertible matrix the O a. Dimension of N(A)=Dn Ob. Dimension of N(A) -Dimension of…

A: From given statement

Q: 2. Suppose f: E" E" is an isometry given by f(v) = Ae for some n x n matrix A, n2 2. a) Show that g:…

A:

Q: Which of the following is true or false? a)Let  A  be nxn matrix. The equality  = is satisfied with…

A: a. Given that A is a n×n matrix. Then by the standard definitions of the inner products we have that…

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2a², –1+x +…

A: "Since you have posted a question with multi subparts, we will solve the first three subparts for…

Q: Exercise 2.2. Given a vector r of appropriate dimension, find the Jacobian F'(x) for each of the…

A:

Q: 2. Let w be a positive continuous function on [0, 1], n be a positive integer, and Pn be the vector…

A: a)Note that the definition of inner product here is completely irrelevant, for any inner product on…

Q: Let B = be any 2 x 2 matrix. COS a (i) Show that there are real numbers un and a such that u11 sin a…

A: Given: A 2×2 matrix, B=abcd

Q: 13) Let V be a complex vector space. Suppose that T is a self-adjoint operator on a inner-product…

A:

Q: = (k₁) be an m x n matrix with scalar entries. For x € K", let Mx € K™ be defined as n Mx(i) = Σkjj…

A:

Q: Let f: C → C be differentiable at zo. Considering f as a map R? → R², that f'(zo) = kR where k is a…

A: (a)

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2x², –1 + x +…

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

Q: 5. Let A be an m x n matrix and define F: R" → R"m by F(r) = Ar. (i) Prove that there exists C > 0…

A: