Question

Prove that the set V of all polynomials of degree ≤ n including the zero polynomial...

Prove that the set V of all polynomials of degree ≤ n including the zero polynomial is vector space over the field R under usual polynomial addition and scalar multiplication. Further, find the basis for the space of polynomial p(x) of degree ≤ 3. Find a basis for the subspace with p(1) = 0.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that...
5. Let S be the set of all polynomials p(x) of degree ≤ 4 such that p(-1)=0. (a) Prove that S is a subspace of the vector space of all polynomials. (b) Find a basis for S. (c) What is the dimension of S? 6. Let ? ⊆ R! be the span of ?1 = (2,1,0,-1), ?2 =(1,2,-6,1), ?3 = (1,0,2,-1) and ? ⊆ R! be the span of ?1 =(1,1,-2,0), ?2 =(3,1,2,-2). Prove that V=W.
If V is a vector space of polynomials of degree n with real numbers as coefficients,...
If V is a vector space of polynomials of degree n with real numbers as coefficients, over R, and W is generated by the polynomials (x 3 + 2x 2 − 2x + 1, x3 + 3x 2 − x + 4, 2x 3 + x 2 − 7x − 7), then is W a subspace of V , and if so, determine its basis.
Let V = Pn(R), the vector space of all polynomials of degree at most n. And...
Let V = Pn(R), the vector space of all polynomials of degree at most n. And let T : V → V be a linear transformation. Prove that there exists a non-zero linear transformation S : V → V such that T ◦ S = 0 (that is, T(S(v)) = 0 for all v ∈ V) if and only if there exists a non-zero vector v ∈ V such that T(v) = 0. Hint: For the backwards direction, consider building...
Suppose V is a vector space over F, dim V = n, let T be a...
Suppose V is a vector space over F, dim V = n, let T be a linear transformation on V. 1. If T has an irreducible characterisctic polynomial over F, prove that {0} and V are the only T-invariant subspaces of V. 2. If the characteristic polynomial of T = g(t) h(t) for some polynomials g(t) and h(t) of degree < n , prove that V has a T-invariant subspace W such that 0 < dim W < n
Prove that the singleton set {0} is a vector subspace of the space P4(R) of all...
Prove that the singleton set {0} is a vector subspace of the space P4(R) of all polynomials of degree at most 3 with real coefficients.
Show that the set GLm,n(R) of all mxn matrices with the usual matrix addition and scalar...
Show that the set GLm,n(R) of all mxn matrices with the usual matrix addition and scalar multiplication is a finite dimensional vector space with dim GLm,n(R) = mn. Show that if V and W be finite dimensional vector spaces with dim V = m and dim W = n, B a basis for V and C a basis for W then hom(V,W)-----MatB--->C(-)--------> GLm,n(R) is a bijective linear transformation. Hence or otherwise, obtain dim hom(V,W). Thank you!
Consider P3 = {a + bx + cx2 + dx3 |a,b,c,d ∈ R}, the set of...
Consider P3 = {a + bx + cx2 + dx3 |a,b,c,d ∈ R}, the set of polynomials of degree at most 3. Let p(x) be an arbitrary element in P3. (a) Show P3 is a vector space. (b) Find a basis and the dimension of P3. (c) Why is the set of polynomials of degree exactly 3 not a vector space? (d) Find a basis for the set of polynomials satisfying p′′(x) = 0, a subspace of P3. (e) Find...
Exercise 9.1.11 Consider the set of all vectors in R2,(x, y) such that x + y...
Exercise 9.1.11 Consider the set of all vectors in R2,(x, y) such that x + y ≥ 0. Let the vector space operations be the usual ones. Is this a vector space? Is it a subspace of R2? Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy = 0. Is this a subspace of R2? Is it a vector space? The addition and scalar multiplication are the usual operations.
Let P4 denote the space of polynomials of degree less than 4 with real coefficients. Show...
Let P4 denote the space of polynomials of degree less than 4 with real coefficients. Show that the standard operations of addition of polynomials, and multiplication of polynomials by a scalar, give P4 the structure of a vector space (over the real numbers R). Your answer should include verification of each of the eight vector space axioms (you may assume the two closure axioms hold for this problem).
use the subspace theorem ( i) is it a non-empty space? ii) is it closed under...
use the subspace theorem ( i) is it a non-empty space? ii) is it closed under vector addition? iii)is it closed under scalar multiplication?) to decide whether the following is a real vector space with its usual operations: the set of all real polonomials of degree exactly n.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT