To find the matrix representation $[T]_{\beta}$ of the linear transformation $T$ with respect to a given basis $\beta$, we need to proceed with the following steps:

Given:

• $T: P_3(\mathbb{R}) \to P_2(\mathbb{R})$ is a linear transformation defined by $T(f) = f'$, where $f'$ is the derivative of $f$.
• The spaces involved:
  • $P_3(\mathbb{R})$ is the space of polynomials of degree at most 3.
  • $P_2(\mathbb{R})$ is the space of polynomials of degree at most 2.

Standard Basis:

Let’s assume the standard basis for both $P_3(\mathbb{R})$ and $P_2(\mathbb{R})$.

• The standard basis for $P_3(\mathbb{R})$ is: $\beta = \{1, x, x^2, x^3\}$

• The standard basis for $P_2(\mathbb{R})$ is: $\beta' = \{1, x, x^2\}$

We now need to find the action of $T(f) = f'$ on each element of the basis for $P_3(\mathbb{R})$, then express the result in terms of the basis for $P_2(\mathbb{R})$.

Step 1: Apply $T$ to the basis elements of $P_3(\mathbb{R})$

1. $T(1) = \frac{d}{dx}(1) = 0$
2. $T(x) = \frac{d}{dx}(x) = 1$
3. $T(x^2) = \frac{d}{dx}(x^2) = 2x$
4. $T(x^3) = \frac{d}{dx}(x^3) = 3x^2$

Step 2: Express results in terms of the basis for $P_2(\mathbb{R})$

Now, express each result as a linear combination of $\{1, x, x^2\}$:

1. $T(1) = 0 = 0 \cdot 1 + 0 \cdot x + 0 \cdot x^2$
2. $T(x) = 1 = 1 \cdot 1 + 0 \cdot x + 0 \cdot x^2$
3. $T(x^2) = 2x = 0 \cdot 1 + 2 \cdot x + 0 \cdot x^2$
4. $T(x^3) = 3x^2 = 0 \cdot 1 + 0 \cdot x + 3 \cdot x^2$

Step 3: Construct the matrix $[T]_{\beta}$

The columns of the matrix $[T]_{\beta}$ correspond to the coefficients of the above results in terms of the basis $\{1, x, x^2\}$. So the matrix will be:

0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$$ ### Final Answer: The matrix representation of the transformation $$T$$ with respect to the standard basis $$\beta$$ is: $$[T]_{\beta} = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$$ Would you like further details or have any questions? --- Here are five related questions to explore this concept further: 1. What would happen if $$T$$ were defined as $$T(f) = f + f'$$? 2. How does the matrix representation change for different bases in $$P_3(\mathbb{R})$$ and $$P_2(\mathbb{R})$$? 3. How can we determine the inverse of the matrix $$[T]_{\beta}$$ if it exists? 4. What is the kernel and image of $$T$$ in this case? 5. How do we represent higher-order derivatives in matrix form? **Tip:** Always verify that the dimension of the matrix matches the dimensions of the spaces involved (here, 3x4 for a map from $$P_3$$ to $$P_2$$).