In problems (1) (2), Pn denotes the set of polynomials of degree at most n with real coef- ficients, on the interval [0, 1], and P denotes the set of all polynomials with real coefficients on the int...
Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5 Let Pn be the set of real polynomials of degree at most n. Show that s (pEP5 :p(-7) (9)) is a subspace of P5
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
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...
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
2. Let P3 stand for the vector space of all polynomials in x with real coefficients and of the degree at most 3. (a) (1 mark) Show that the set E = {p(x) € P3 : p(3)=0}, is a subspace of P3. (b) (2 marks) Show that the collection of polynomials {(x - 3), (x – 3), (x-3)3} is a basis of E.
TL 0o Pn P, an 7-0 (a) Define T : Pl → R2 by T(ao + air) = (ao, (n). Note that T (2) L(P..R2). Fin(! (b) Define T1 R2P by T-(ao, aa. Note that TE L(R2.P). Find IT-1 TL 0o Pn P, an 7-0 (a) Define T : Pl → R2 by T(ao + air) = (ao, (n). Note that T (2) L(P..R2). Fin(! (b) Define T1 R2P by T-(ao, aa. Note that TE L(R2.P). Find IT-1
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...
Could a convolutional net learn calculus? Start with the derivatives of fourth degree polynomials p(x). The inputs could be graphs of p o + aix + … + a4-for 0 S a S 1 and a training set of a's. The correct outputs would be the coefficients 0, ai , 2аг, Заз, 4a4 from dp/dz. Using softmax with 5 classes, could you design and create a CNN to learn differential calculus ? Could a convolutional net learn calculus? Start with...
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P given by g) = Lif(t)g(x).rº dr. (a) Prove that (,) defines an inner product on P. (b) Let O be the set of odd polynomials, i.e. f(r) € P such that f(x)= -f(-r). Show that is a subspace of P. (c) Explain why g() = 5x2 - 3 is in 0+ (the orthogonal complement of O with respect to (>). (d) Let P<2 denote...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...