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.

Learn how to express T(2x) as a polynomial with coefficients in terms of a and b. This problem involves linear operators and basis representation, with a detailed step-by-step solution using matrix representations and coefficients. Suitable for advanced undergraduate students studying linear algebra and transformations.

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