Let's tackle each part of this problem step-by-step. Here is a breakdown of the solution approach:

Problem

You are working in the vector space $\mathbb{R}_2[x]$, the space of polynomials of degree at most 2, and you have a mapping $A: \mathbb{R}_2[x] \rightarrow \mathbb{R}_2[x]$ defined by: $A(p)(x) := p(x) + (x + 1)p'(x)$ where $p'(x)$ is the derivative of $p(x)$.

Solution

Part (a): Show that $A$ is a linear operator.

To show $A$ is a linear operator, we must demonstrate that it satisfies linearity, specifically:

1. $A(p + q) = A(p) + A(q)$ for any polynomials $p$ and $q$ in $\mathbb{R}_2[x]$,
2. $A(cp) = cA(p)$ for any polynomial $p \in \mathbb{R}_2[x]$ and scalar $c$.

Since $A$ involves both $p(x)$ and $p'(x)$, you can show each property holds by applying the definition of $A$ and the properties of derivatives.

Part (b): Find the matrix of $A$ with respect to the basis $\mathcal{B} = \{1, x, x^2\}$.

To find the matrix of $A$ relative to the basis $\mathcal{B} = \{1, x, x^2\}$, apply $A$ to each of the basis elements and express the results in terms of the basis.

1. Apply $A$ to $1$:

   • $p(x) = 1 \Rightarrow p'(x) = 0$.
   • Thus, $A(1)(x) = 1$.

2. Apply $A$ to $x$:

   • $p(x) = x \Rightarrow p'(x) = 1$.
   • $A(x) = x + (x + 1) \cdot 1 = 2x + 1$.

3. Apply $A$ to $x^2$:

   • $p(x) = x^2 \Rightarrow p'(x) = 2x$.
   • $A(x^2) = x^2 + (x + 1) \cdot 2x = x^2 + 2x^2 + 2x = 3x^2 + 2x$.

Now, express each result as a linear combination of $\{1, x, x^2\}$ to construct the matrix for $A$.

Part (c): Find the matrix of $A$ with respect to the basis $\mathcal{R} = \{1, (x - 1), (x - 1)^2\}$.

Follow a similar approach, but this time with the basis $\mathcal{R} = \{1, (x - 1), (x - 1)^2\}$. Apply $A$ to each basis element, express the result in terms of $\{1, (x - 1), (x - 1)^2\}$, and construct the matrix.

Part (d): Find the polynomials for which $A(p)(x) = 0$.

To find the polynomials $p(x)$ that satisfy $A(p)(x) = 0$, set $A(p)(x) = p(x) + (x + 1)p'(x) = 0$ and solve this differential equation for $p(x)$ in $\mathbb{R}_2[x]$.

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Would you like to go through each part in more detail, including specific calculations for each matrix and the steps to show linearity? Let me know if you have questions on any part, or if you want me to proceed with explicit calculations.

Relative Questions

1. What steps are involved in proving that a mapping is a linear operator?
2. How do we express a polynomial as a linear combination of basis elements?
3. What are the differences between solving for matrices in different bases?
4. What methods can we use to solve differential equations for polynomials?
5. How does changing the basis affect the representation of a linear transformation?

Tip

When finding matrices for linear transformations in different bases, always apply the transformation to each basis element and express the results in terms of the chosen basis—this keeps the process systematic and clear.