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Question 4) (6 points) Below are two unrelated questions that both deal with inner products. Note...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
How to solve all of this linear Algebra 8. (24 points total) LetV be the vector space{P2, +, *}with standard function addition and scalar multiplication Define an Inner product: <p | q>= p(0)q[O) + p(1)q(1)+ p(2)q(2). Let B = {x,x,1} a. Explain why this inner product satisfies the positive property b. Explain how you know that B forms a basis c. State the conclusions of Cauchy-Schwartz and the Triangle inequalities in terms of this inner product d. Use Gram-Schmidt and...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
NEED (B) AND (C) 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space C(I-1,1) of continuous real-valued funo- tions on the domain [-1, 1] (b) Use the Gram-Schmidt process to find an orthonormal basis for P2(R) with re- spect to this inner product (c) Find a polynomial q(x) such that for every p E P2R 2. (a) Prove that 1 (f, g)=| x2 f(x)g(x)dx is an inner product on the vector space...
Question 10 (6 marks) Consider the inner product on P2 defined by rl 0 Find two linearly independent polynomials that are orthogonal to p(x)-1+r.
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v, w, u} are linearly independent vectors in some vector space V. Prove then that the vectors {v + w, w + u,v + u} are also linearly independent in V. (b) Suppose T is a linear transformation, T: P10(R) → M3(R) Prove that T cannot be 1-to-1 (c) Prove that in ANY inner product that if u and w are unit vectors (ie ||vl|...
Please attempt both questions 3. Use the continuous function on the interval (0,1) inner product to find the projection of f(x) onto g(x). (Feel free to use an integral calculator. I use wolfram alpha. Just make sure to type the problem in carefully). (a) f(x) = -22 - 1, g(x) = -2 (b) f(x) = 2r?, g(x) = 2+1 (e) f(x)=-1-1, g(x) = x2 +3 4. Consider 3-space with the dot product. Your subspace S will be the plane z...
Please help for Question 10A.1 MATH 270 SPRING 2019 HOMEWORK 10 10A. 1. Let S be the subspace in R3 spanned by21.Find a basis for S 2. Using as the inner product (5) ( p. 246) in section 5.4 for Ps where x10, x2 -1, x3 - 2: Find the angle between p (x) = x-3 and q(x) = x2 + x + 2. b. Fnd the vector projection of p(x) on q(x) In Cl-π, π} using as an inner...
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [ 3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ] (i) Calculate A + B, (ii) Calculate AB (iii) Calculate the inverse of B, (iv) Calculate the determinant of C. (b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2) and (5, −1, 4) respectively. (i) Show that P Q~ =...
Question 3: (a) (4 points) Recall that the trace of a square matrix is the sum of all its entries from the main diagonal. Show that the trace is linear, in the sense that, trace(aX + βΥ) trace(X) + β trace(Y). Let V be the space of all m × n matrices. A function <..) : V × V → R is defined as (A, B) trace(ABT), A, B E V. (a) (4 points) Using the properties of the trace,...