3. [20 points] Recall the vector space of polynomials of degree up to 2, (P2, +,); see Example 2.4.2 in the book. Consider the sets W₁ = {p € P2 | p(x) = ax² + bx +c, W₂ = {p € P2 | p(x) = ar² + bx + c, a = c = c}, a = 2c = 2c}. (a) Prove if W₁ UW2 a subspace of P2. (b) Prove if W₁n W₂ a subspace of P2. (c) Prove if W₁ + W2 a subspace of P2. Hint: Use the applicable theorems covered in the book, section 2.5.4. Example 2.4.2 Let Pn (R) be the set of all polynomials of degree less than or equal to n with coefficients from R. That is, Pn(R) = {ao+a₁x + ……… · +ɑnx” | ak Є R, k = 0, 1, … · ·‚n} . Let f(x) = ao+a₁x + ··· + anx" and g(x) = bo+b₁x + ··· + b₁x" be polynomials in Pn (R) and a = R. Define addition and scalar multiplication component-wise: (f+g)(x) = (ao+bo) + (a₁ + b₁)x + ... + (an+bn)x", (a f)(x) = (aa0) + (aa₁)x + ··· + (aan)x".
3. [20 points] Recall the vector space of polynomials of degree up to 2, (P2, +,); see Example 2.4.2 in the book. Consider the sets W₁ = {p € P2 | p(x) = ax² + bx +c, W₂ = {p € P2 | p(x) = ar² + bx + c, a = c = c}, a = 2c = 2c}. (a) Prove if W₁ UW2 a subspace of P2. (b) Prove if W₁n W₂ a subspace of P2. (c) Prove if W₁ + W2 a subspace of P2. Hint: Use the applicable theorems covered in the book, section 2.5.4. Example 2.4.2 Let Pn (R) be the set of all polynomials of degree less than or equal to n with coefficients from R. That is, Pn(R) = {ao+a₁x + ……… · +ɑnx” | ak Є R, k = 0, 1, … · ·‚n} . Let f(x) = ao+a₁x + ··· + anx" and g(x) = bo+b₁x + ··· + b₁x" be polynomials in Pn (R) and a = R. Define addition and scalar multiplication component-wise: (f+g)(x) = (ao+bo) + (a₁ + b₁)x + ... + (an+bn)x", (a f)(x) = (aa0) + (aa₁)x + ··· + (aan)x".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3. [20 points]
Recall the vector space of polynomials of degree up to 2, (P2, +,); see Example 2.4.2 in the book.
Consider the sets
W₁ = {p € P2 | p(x) = ax² + bx +c,
W₂ = {p € P2 | p(x) = ar² + bx + c,
a
= c
= c},
a
= 2c
= 2c}.
(a) Prove if W₁ UW2 a subspace of P2.
(b) Prove if W₁n W₂ a subspace of P2.
(c) Prove if W₁ + W2 a subspace of P2.
Hint: Use the applicable theorems covered in the book, section 2.5.4.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9328926a-b343-4d47-8a00-cbd9e6e36ac2%2F746b4cf3-71b4-4ca5-aacd-f503ae66b810%2Fkfkhmmm_processed.png&w=3840&q=75)
Transcribed Image Text:3. [20 points]
Recall the vector space of polynomials of degree up to 2, (P2, +,); see Example 2.4.2 in the book.
Consider the sets
W₁ = {p € P2 | p(x) = ax² + bx +c,
W₂ = {p € P2 | p(x) = ar² + bx + c,
a
= c
= c},
a
= 2c
= 2c}.
(a) Prove if W₁ UW2 a subspace of P2.
(b) Prove if W₁n W₂ a subspace of P2.
(c) Prove if W₁ + W2 a subspace of P2.
Hint: Use the applicable theorems covered in the book, section 2.5.4.
Transcribed Image Text:Example 2.4.2 Let Pn (R) be the set of all polynomials of degree less than or equal to n with
coefficients from R. That is,
Pn(R) =
{ao+a₁x + ……… · +ɑnx” | ak Є R, k = 0, 1, … · ·‚n} .
Let f(x) = ao+a₁x + ··· + anx" and g(x) = bo+b₁x + ··· + b₁x" be polynomials in Pn (R) and
a = R. Define addition and scalar multiplication component-wise:
(f+g)(x) = (ao+bo) + (a₁ + b₁)x + ... + (an+bn)x",
(a f)(x) = (aa0) + (aa₁)x + ··· + (aan)x".
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