Question

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 a basis for the subspace of P3 consisting of the polynomials with p(1) = 0.

Math   Algebra   
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
Let the set W be: all polynomials in P3 satisfying that p(-t)=p(t), Question: Is W a...
Let the set W be: all polynomials in P3 satisfying that p(-t)=p(t), Question: Is W a vector space or not? If yes, find a basis and dimension
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.
In this question we denote by P2(R) the set of functions {ax2 + bx + c...
In this question we denote by P2(R) the set of functions {ax2 + bx + c : a, b, c ∈ R}, which is a vector space under the usual addition and scalar multiplication of functions. Let p1, p2, p3 ∈ P2(R) be given by p1(x) = 1, p2(x) = x + 2x 2 , and p3(x) = αx + 4x 2 . a) Find the condition on α ∈ R that ensures that {p1, p2, p3} is a basis...
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.
Define T : P2 → R3 via T(a+bx+cx2) = (a+c,c,b−c), and let B = {1,x,x2} and...
Define T : P2 → R3 via T(a+bx+cx2) = (a+c,c,b−c), and let B = {1,x,x2} and D ={(1, 0, 0), (0, 1, 0), (0, 0, 1)}. (a) Find MDB(T) and show that it is invertible. (b) Use the fact that MBD(T−1) = (MDB(T))−1 to find T−1. Hint: A linear transformation is completely determined by its action on any spanning set and hence on any basis.
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.
Question 4. Consider the following subsets of the vector space P3 of polynomials of degree 3...
Question 4. Consider the following subsets of the vector space P3 of polynomials of degree 3 or less: S = {p(x) : p(1) = 0} and T = {q(x) : q(0) = 1} Determine if these subsets are vectors spaces with the standard operations for polynomials
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2,...
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2, and ?S is the subset of ?2P2 consisting of all polynomials of the form ?(?)=?2+?.p(x)=x2+c. B. ?=?5(?)V=C5(I), and ?S is the subset of ?V consisting of those functions satisfying the differential equation ?(5)=0.y(5)=0. C. ?V is the vector space of all real-valued functions defined on the interval [?,?][a,b], and ?S is the subset of ?V consisting of those functions satisfying ?(?)=?(?).f(a)=f(b). D. ?=?3(?)V=C3(I), and...
Let F be a field (for instance R or C), and let P2(F) be the set...
Let F be a field (for instance R or C), and let P2(F) be the set of polynomials of degree ≤ 2 with coefficients in F, i.e., P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}. Prove that P2(F) is a vector space over F with sum ⊕ and scalar multiplication defined as follows: (a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x + (a2 + b2)x^2 λ (b0 +...
We have learned that we can consider spaces of matrices, polynomials or functions as vector spaces....
We have learned that we can consider spaces of matrices, polynomials or functions as vector spaces. For the following examples, use the definition of subspace to determine whether the set in question is a subspace or not (for the given vector space), and why. 1. The set M1 of 2×2 matrices with real entries such that all entries of their diagonal are equal. That is, all 2 × 2 matrices of the form: A = a b c a 2....
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT