Question

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.
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
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.
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
Let P2 denote the vector space of polynomials in x with real coefficients having degree at...
Let P2 denote the vector space of polynomials in x with real coefficients having degree at most 2. Let W be a subspace of P2 given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper subspace of P2.
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...
For a nonempty subset S of a vector space V , define span(S) as the set...
For a nonempty subset S of a vector space V , define span(S) as the set of all linear combinations of vectors in S. (a) Prove that span(S) is a subspace of V . (b) Prove that span(S) is the intersection of all subspaces that contain S, and con- clude that span(S) is the smallest subspace containing S. Hint: let W be the intersection of all subspaces containing S and show W = span(S). (c) What is the smallest subspace...
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 ℙn be the set of real polynomials of degree at most n, and write p′...
Let ℙn be the set of real polynomials of degree at most n, and write p′ for the derivative of p. Show that S={p∈ℙ9:p(2)=−1p′(2)} is a subspace of ℙ9.
Let P be the vector space of all polynomials in x with real coefficients. Does P...
Let P be the vector space of all polynomials in x with real coefficients. Does P have a basis? Prove your answer.
4. Prove the Following: a. Prove that if V is a vector space with subspace W...
4. Prove the Following: a. Prove that if V is a vector space with subspace W ⊂ V, and if U ⊂ W is a subspace of the vector space W, then U is also a subspace of V b. Given span of a finite collection of vectors {v1, . . . , vn} ⊂ V as follows: Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in the scalar field}...
Let V be the set of polynomials of the form ax + (a^2)(x^2), for all real...
Let V be the set of polynomials of the form ax + (a^2)(x^2), for all real numbers a. Is V a subspace of P?
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT