To express $T(2x)$ as a polynomial with coefficients in terms of $a$ and $b$, let's follow these steps:

Step 1: Express $2x$ in the Basis $\mathcal{B}$

We know that: $2x = c_1(1 - 3x) + c_2(1 + 3x)$ Let's solve this system:

1. Expand the right-hand side: $2x = c_1 - 3c_1x + c_2 + 3c_2x$ $2x = (c_1 + c_2) + (-3c_1 + 3c_2)x$

2. Match coefficients of like terms:

   • Coefficient of $x$: $2 = -3c_1 + 3c_2$
   • Constant term: $0 = c_1 + c_2$

   Solving the system:

   • From $c_1 + c_2 = 0$, we get $c_2 = -c_1$.
   • Substitute into the equation for the $x$ term: $2 = -3c_1 + 3(-c_1)$ $2 = -6c_1$ $c_1 = -\frac{1}{3}, \quad c_2 = \frac{1}{3}$

Thus, we have: $2x = -\frac{1}{3}(1 - 3x) + \frac{1}{3}(1 + 3x)$

Step 2: Apply the Linear Operator $T$ to $2x$

Given the matrix $[T]_{\mathcal{B}} = \begin{pmatrix} 1 & 2 \\ -3 & 0 \end{pmatrix}$, we can now express $T(2x)$ using the previously found coefficients:

$T(2x) = T\left(-\frac{1}{3} b_1 + \frac{1}{3} b_2 \right)$

This can be written as: $T(2x) = -\frac{1}{3} T(b_1) + \frac{1}{3} T(b_2)$

Using the matrix $[T]_{\mathcal{B}}$: $T(b_1) = b_1 + 2b_2 = (1 - 3x) + 2(1 + 3x) = 3 + 3x$ $T(b_2) = -3b_1 = -3(1 - 3x) = -3 + 9x$

Thus: $T(2x) = -\frac{1}{3}(3 + 3x) + \frac{1}{3}(-3 + 9x)$

Step 3: Simplify and Express in Terms of $a$ and $b$

Simplify: $T(2x) = -1 - x + (-1 + 3x)$ $T(2x) = -2 + 2x$

Finally, express $T(2x)$ as $ax + b$, where:

• $a = 2$
• $b = -2$

Thus: $T(2x) = 2x - 2$

Conclusion:

$T(2x) = 2x - 2$ with coefficients $a = 2$ and $b = -2$.

Would you like further details or have any questions about this problem?

Next Steps:

1. Can you interpret the result $T(2x) = 2x - 2$ in terms of linear transformations?
2. How does changing the basis $\mathcal{B}$ affect the coefficients $a$ and $b$?
3. How would you find $T(p(x))$ for a general polynomial $p(x) = ax + b$?
4. What if the basis vectors were different—how would the result change?
5. Can you verify the matrix representation of $T$ in both bases $\mathcal{E}$ and $\mathcal{B}$?
6. How does this problem relate to eigenvalues and eigenvectors?
7. Could you express $T$ using another set of polynomials, such as the standard monomials?
8. How would the matrix $[T]_{\mathcal{E}}$ change if you swapped the basis vectors?

Tip: Working through problems like this can help solidify your understanding of linear operators and their representations in different bases.