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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".

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 45E
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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: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".
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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