Subspace Proof & Basis For W In P3

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Subspace Proof & Basis for W in P3

Let's dive into proving that a given set W is a subspace of the vector space P3, and then figure out how to find a basis for it. This is a common task in linear algebra, and understanding the process is super important. So, let's break it down step by step.

Proving W is a Subspace of P3

To show that W is a subspace of P3, where P3 is the vector space of all polynomials of degree at most 3, we need to verify three conditions:

  1. The zero vector is in W: The zero vector in P3 is the zero polynomial, p(x) = 0. We need to check if this polynomial satisfies the condition that defines W. This is often the easiest condition to check. Make sure that when all coefficients are zero, the condition holds true. If the zero vector isn't in W, then W is not a subspace, and you can stop there.
  2. Closure under addition: If p(x) and q(x) are both in W, then their sum, (p+q)(x) = p(x) + q(x), must also be in W. This means that if p(x) and q(x) satisfy the condition for membership in W, then their sum must also satisfy that condition. This is where you'll likely use the properties of polynomials and algebraic manipulation to demonstrate that the sum still meets the required criteria. For instance, if W consists of polynomials p(x) such that p(0) = 0, then if p(0) = 0 and q(0) = 0, it must be shown that (p+q)(0) = 0.
  3. Closure under scalar multiplication: If p(x) is in W, then for any scalar c, the scalar multiple cp(x) must also be in W. This requires showing that if p(x) satisfies the condition for being in W, then cp(x) also satisfies that condition. Similar to closure under addition, this often involves using the properties of polynomials and scalar multiplication to confirm that the scalar multiple still meets the necessary criteria. If W consists of polynomials p(x) such that p(0) = 0, then if p(0) = 0, it must be shown that cp(0) = 0 for any scalar c.

If all three of these conditions are met, then W is indeed a subspace of P3. Guys, remember that showing these conditions rigorously is key to a solid proof.

Let's illustrate with an example. Suppose W is the set of all polynomials p(x) in P3 such that p(0) = 0. That is, W = {p(x) ∈ P3 | p(0) = 0}.

  1. Zero vector: The zero polynomial p(x) = 0 satisfies p(0) = 0, so the zero vector is in W.
  2. Closure under addition: Let p(x) and q(x) be in W, so p(0) = 0 and q(0) = 0. Then (p+q)(0) = p(0) + q(0) = 0 + 0 = 0, so p(x) + q(x) is in W.
  3. Closure under scalar multiplication: Let p(x) be in W, so p(0) = 0. For any scalar c, (cp)(0) = cp(0) = c0 = 0*, so cp(x) is in W.

Since all three conditions are satisfied, W is a subspace of P3.

Finding a Basis for W

Once we've established that W is a subspace, the next step is to find a basis for it. A basis is a set of linearly independent vectors that span the subspace. In other words, every vector in the subspace can be written as a linear combination of the basis vectors.

Here’s how you can find a basis for W:

  1. Express the general form of polynomials in W: Start by expressing a general polynomial p(x) in P3 as p(x) = a + bx + cx^2 + dx^3, where a, b, c, and d are scalars. Then, apply the condition that defines W to this general form. This will give you a relationship between the coefficients.
  2. Use the condition to eliminate variables: Use the condition that defines W to express some of the coefficients in terms of the others. This reduces the number of independent variables and helps you express the general polynomial in a simpler form. For example, if W is defined by polynomials p(x) such that p(0) = 0, then a = 0, and the polynomial simplifies to p(x) = bx + cx^2 + dx^3.
  3. Write the general polynomial as a linear combination: Express the general polynomial as a linear combination of linearly independent polynomials. The coefficients of these polynomials will be the remaining independent variables. These linearly independent polynomials will form a basis for W.
  4. Verify linear independence and spanning: Ensure that the set of polynomials you've identified is linearly independent and spans W. Linear independence means that none of the polynomials can be written as a linear combination of the others. Spanning means that every polynomial in W can be written as a linear combination of the basis polynomials. This step confirms that you've found a valid basis.

Continuing with our example where W = {p(x) ∈ P3 | p(0) = 0}, we found that p(x) = bx + cx^2 + dx^3. We can write this as:

p(x) = b(x) + c(x^2) + d(x^3)

Here, b, c, and d are scalars, and the polynomials x, x^2, and x^3 are linearly independent. Thus, a basis for W is {x, x^2, x^3}

To verify linear independence, suppose we have:

k1x + k2x^2 + k3x^3 = 0

for all x, where k1, k2, and k3 are scalars. This equation must hold for all x, which implies that k1 = k2 = k3 = 0. Therefore, the polynomials are linearly independent.

To verify that they span W, note that any polynomial p(x) in W can be written in the form bx + cx^2 + dx^3, which is a linear combination of x, x^2, and x^3. Thus, {x, x^2, x^3} is indeed a basis for W.

Common Pitfalls and How to Avoid Them

  • Forgetting to check all three conditions for a subspace: It’s crucial to verify the zero vector condition, closure under addition, and closure under scalar multiplication. Skipping any of these steps can lead to incorrect conclusions.
  • Assuming linear independence without proof: Always verify that the polynomials you've identified as a basis are indeed linearly independent. Use the definition of linear independence to prove it.
  • Not expressing the general form correctly: Make sure you express the general form of the polynomials in P3 correctly before applying the conditions that define W. An incorrect general form can lead to an incorrect basis.
  • Algebraic errors: Be careful with algebraic manipulations when verifying closure under addition and scalar multiplication. Simple errors can invalidate your proof.

Practical Tips for Success

  • Start with the zero vector: Always check if the zero vector is in W first. If it’s not, then you know immediately that W is not a subspace.
  • Use clear notation: Use clear and consistent notation to avoid confusion. This is particularly important when dealing with polynomials and scalars.
  • Practice with examples: Work through several examples to solidify your understanding. This will help you become more comfortable with the process and identify common patterns.
  • Double-check your work: Always double-check your work, especially the algebraic manipulations. A small error can lead to a wrong answer.

Conclusion

Proving that W is a subspace of P3 and finding a basis for W involves verifying three key conditions and then expressing the general polynomial in W as a linear combination of linearly independent polynomials. By following these steps carefully and avoiding common pitfalls, you can confidently tackle these types of problems. Keep practicing, and you'll become a pro in no time! Remember that linear algebra is like building with LEGOs – each concept builds on the previous one. Good luck, guys!