Check that the subset {p(x) = a0 + a1x + a2x2 + a3x3|p'(0) = 0} of...
Check that the subset {p(x) = a0 + a1x + a2x2 + a3x3|p'(0) = 0} of degree at most three polynomials whose derivative at 0 is 0 is a vector space. What would be a basis for it?
Homework Answers
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Check that the subset {p(x) = a0 + a1x +
a2x2 + a3x3|p'(0) = 0}
of...
6. Let Pm (F) be the vector space of polynomials p(x) = ao + a1x +...
6. Let Pm (F) be the vector space of polynomials p(x) = ao + a1x + ... Amx" with coefficients in F and degree at most m, and let U be the set of even polynomials in P5(F): U := {p(x) € P5(F) | P(x) = p(-x)}. (a) Show that the list of vectors 1, x, x², x3, x4 + x, x + x spans P5(F). (b) Show that U is a vector subspace of P5(F) (c) Prove that there...
Show how to apply the fast Fourier Transform to multiply three polynomials A(x) = a0 +a1x+a2x...
Show how to apply the fast Fourier Transform to multiply three polynomials A(x) = a0 +a1x+a2x 2 +· · ·+an−1x n−1 of degree n−1, B(x) = b0 +b1x+b2x 2 +· · ·+bn−1x n−1 of degree n − 1, and C(x) = c0 + c1x + c2x 2 + · · · + cn−1x n−1 of degree n − 1 when n is a power of 2. Analyze its time complexity.
Q3. Recall that P, is the vector space of all real polynomials of degree at most...
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p' (t)=0}, where p' (t) is the derivative of the polynomial p(t).
Q3. Recall that P, is the vector space of all real polynomials of degree at most...
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p'(t)=0}, where p' (t) is the derivative of the polynomial p(t).
Let V be the subset of P that consists of polynomials in X who have a...
Let V be the subset of P that consists of polynomials in X who have a second derivative equal to 0. V = = {pl) in P such that de: (a) = 0} = {p(x) in P such that p"(x) = 0; Show that V is or is not a subspace of P.
The set of polynomials p(x) = ax2 + bx + c that satisfy p(3) = 0...
The set of polynomials p(x) = ax2 + bx + c that satisfy p(3) = 0 is a subspace of the vector space P2 of all polynomials of degree two or less. O True False
6. For the following vector spaces V, determine if the subset H is a subspace. If...
6. For the following vector spaces V, determine if the subset H is a subspace. If not, give one reason why H fails to be a subspace. (a) (5 points) V is the set of functions f from R + R, and H is the set of polynomials of integer coefficients. (b) (5 points) V = P, is the vector space of polynomials of degree at most 2, and H is the subset of all polynomials in P2 of the...
Hi there, I literally got stuck on this question, it would be great if someone can give me help, ...
Hi there, I literally got stuck on this question, it would be
great if someone can give me help, many thanks in advance!
A polynomial on R is a function p : R -R of the form p(x) - aj' where each ai E R and at most finitely many ai are nonzero. Let P denote the set of all polynomials on R (a) What is the dimension of P, regarded as a vector space over R? You do not...
Please provide answer in neat handwriting. Thank you Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operato...
Please provide answer in neat handwriting. Thank you
Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
2. Let P3 stand for the vector space of all polynomials in x with real coefficients...
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.
6. Let Pm (F) be the vector space of polynomials p(x) = ao + a1x + ... Amx" with coefficients in F and degree at most m, and let U be the set of even polynomials in P5(F): U := {p(x) € P5(F) | P(x) = p(-x)}. (a) Show that the list of vectors 1, x, x², x3, x4 + x, x + x spans P5(F). (b) Show that U is a vector subspace of P5(F) (c) Prove that there...
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p' (t)=0}, where p' (t) is the derivative of the polynomial p(t).
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p'(t)=0}, where p' (t) is the derivative of the polynomial p(t).
Let V be the subset of P that consists of polynomials in X who have a second derivative equal to 0. V = = {pl) in P such that de: (a) = 0} = {p(x) in P such that p"(x) = 0; Show that V is or is not a subspace of P.
The set of polynomials p(x) = ax2 + bx + c that satisfy p(3) = 0 is a subspace of the vector space P2 of all polynomials of degree two or less. O True False
6. For the following vector spaces V, determine if the subset H is a subspace. If not, give one reason why H fails to be a subspace. (a) (5 points) V is the set of functions f from R + R, and H is the set of polynomials of integer coefficients. (b) (5 points) V = P, is the vector space of polynomials of degree at most 2, and H is the subset of all polynomials in P2 of the...
Hi there, I literally got stuck on this question, it would be
great if someone can give me help, many thanks in advance!
A polynomial on R is a function p : R -R of the form p(x) - aj' where each ai E R and at most finitely many ai are nonzero. Let P denote the set of all polynomials on R (a) What is the dimension of P, regarded as a vector space over R? You do not...
Please provide answer in neat handwriting. Thank you
Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
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.
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