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6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R} Consider the following collections of functions from P3(R). W₁ = {ƒ Є P3(R) | ƒ(1) = 0 and ƒ(0) = 1}, W₂ = {ƒ Є P3(R) | ƒ(−1) = ƒ(0) and ƒ(1) = ƒ(0)} a) One of W₁ and W₂ is not a subspace of P3(R). Which one, and why not? b) One of W₁ and W2 is a subspace of P3(R). Which one? Prove it. c) Find a basis for the subspace that you found in part (b), and find its dimension.

6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R} Consider the following collections of functions from P3(R). W₁ = {ƒ Є P3(R) | ƒ(1) = 0 and ƒ(0) = 1}, W₂ = {ƒ Є P3(R) | ƒ(−1) = ƒ(0) and ƒ(1) = ƒ(0)} a) One of W₁ and W₂ is not a subspace of P3(R). Which one, and why not? b) One of W₁ and W2 is a subspace of P3(R). Which one? Prove it. c) Find a basis for the subspace that you found in part (b), and find its dimension.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R} Consider the following collections of functions from P3(R). W₁ = {ƒ Є P3(R) | ƒ(1) = 0 and ƒ(0) = 1}, W₂ = {ƒ Є P3(R) | ƒ(−1) = ƒ(0) and ƒ(1) = ƒ(0)} a) One of W₁ and W₂ is not a subspace of P3(R). Which one, and why not? b) One of W₁ and W2 is a subspace of P3(R). Which one? Prove it. c) Find a basis for the subspace that you found in part (b), and find its dimension.
Transcribed Image Text:6. (Basis and Dimension.) Say P3(R) = {f: R → R | f(x) = ax³ + bx²+cx+d for some a, b, c, d = R} Consider the following collections of functions from P3(R). W₁ = {ƒ Є P3(R) | ƒ(1) = 0 and ƒ(0) = 1}, W₂ = {ƒ Є P3(R) | ƒ(−1) = ƒ(0) and ƒ(1) = ƒ(0)} a) One of W₁ and W₂ is not a subspace of P3(R). Which one, and why not? b) One of W₁ and W2 is a subspace of P3(R). Which one? Prove it. c) Find a basis for the subspace that you found in part (b), and find its dimension.
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