E the vector space of polynomials over R of degree less than 3. Define a quadratic form on V by a...
7. Let V = Pa(R), the vector space of polynomials over R of degree less than 2, with inner product Define φ E p by φ(g)-g(-1) a) By direct calculation, find f e V such that (S)-dg). You are given that A 1, V3-2v) is an orthonormal basis for V (you do not need to check this). b) Find the same f as in part a, using the formula for A(6) from class. 7. Let V = Pa(R), the vector...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements 14. Let Ps denote the vector space of polynomials of degree 3 or less,...
(1 point) Let Ps be the vector space of all polynomials of degree at most 3, and consider the subspace 11 = {r(z) e Pal p(1) = 0} of P3 a A basis for the subspace H is { 22x+12x^2-x-1 Enter your answer as a comma separated list of polynomials. b. The dimension of His 3 (1 point) Find a basis for the space of symmetric 2 x 2-matrices If you need fewer basis elements than there are blanks provided,...
(1 point) Let V be the vector space P3[x] of polynomials in x with degree less than 3 and W be the subspace a. Find a nonzero polynomial p(x) in W b. Find a polynomial q(x) in V\ W. q(x)-
3. P. is the vector space of all polynomials of degree n or less and the zero polynomial Define a derivative transformation T as follow: T. +P, T(+241 +0,2%) = 41 + 2121 (a) (10 Puan) Find the corresponding matrix for T. (b) (10 Puan) Choose your polynomial in P, and find the derivative of your polynomial by using the matrix in (a).
2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг + dr + ey + f | a,b,c, d, e, f E R} Let T be the linear operator on V defined by Find the Jordan canonical form of T 2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг...
Q2 (10 points) Let V- Ps be the vector space of polynomials of degree 3. Let C (1,x 2, 2)2 +2)3) be two ordered bases of V. () Find the change-of-basis matrices Pc-B and PB-c (ii) Find [y]в if [v]c- (1, 0, 0, 1). (iii) Find [y]c if [y]B-( 1, 0, 0, 1).
C- haCh 6 Recall that Ps is the vector space of polynomials with degree less than 3 ay (6 points) Show that (x,x-1,2+1) is a spanning set of Ps (that is, any quadratic polynomial ar2+ bz + c is a linear combination of r, r -1, and ? +1). (b) (6 points) Show that , z-1,ェ2 + 1 are linearly independent. (c) (2 points) What do parts (a) and (b) show about the dimension of P? 0N t u Spanning...
Let V be the vector space of all polynomials of degree at most 2 equipped with the inner product defined by (p,q) = p(-1)q (-1) + p (0)g(0) +p(1)q(1),p(x),g(x) E V Find a nonzero polynomial that is orthogonal to both p(x) = 1 + x + x2, and q(x) = 1-2x + x2
Let V = R3[x] be the vector space of all polynomials with real coefficients and degress not exceeding 3. Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...