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QUESTION 1 Vn n Yn Define L: R"X" → R"X" by L(A)=2A , Ais matrix ER"X"...
QUESTION 3 Define (u,v) = 2q + b2 for u =(Q4,22,23), v = (b4.62,63) ER is it inner product. (explain) Attach File Browse My Computer
Prove Lemma a) Fix a basis {v1, v2, . . . , vn} for an n-dimensional vector space V. Define a linear operator T : V → Fn in the following way: For each x = Σni=1 civi ∈ V, define . Then T is a linear operator. b) Let T be a linear operator from V to W. Suppose that {v1, v2, . . . , vn} is a basis for V and {w1, w2, . . . ,...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
QUESTION 2 Let C be the curve represented by the parametric equation x(t) = ? +61 +12t+36 yt=+1 Find an equation for the tangent line at the point t-1 on the curve C. Where on the curve is the tangent line vertical? Attach File Browse My Computer QUESTION 3 Express the Cartesian coordinates (-1,1) in polar coordinates in at least two different ways. Attach File Browse My Computer
Let I, Y ER" be two nonzero n-dimensional vectors and define the n x n matrix A = ty eigenvalues of A are 0 and y's Show that the
Question 2 It is known that Fourier series of f(x) = x is 2 -% 26 – 1)" * * sin(nx) on interval [- TT, TT). Use this to find the value of the infinite sum 1 - 1 + 3 5 7 Attach File Browse My Computer for Copyright Cleared File Browse Content Coection
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...
5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x)) = p(t). Here we consider R as the vector space R1. (a) Prove evt is a linear map. (b) For part (b), let n= 4. Write down a polynomial p e ker(ev3). (c) For any t, the set of polynomials Ut = {p E Pn(R) : p(t) = 0} is a subspace. What is the dimension of Ut (in terms of n)? Justify your...