Use Theorem 4.3.2 to find the matrix B of the linear transformation [latex]T ( f ) = f^{\prime} + f^{\prime \prime}[/latex] from [latex]P_2[/latex] to [latex]P_2[/latex] with respect to the standard basis [latex]𝔅 = (1, x, x^2)[/latex]; see Example 1.

By Theorem 4.3.2, we have [latex]B=\begin{bmatrix}[T(1)]_𝔅&[T(x)]_𝔅&[T(x^2)]_𝔅\end{bmatrix}[/latex]. Now As in Section 3.4, it is suggestive to write the basis elements 1, x, and [latex]x^2[/latex] next to the rows of matrix B, while the values T (1) = 0, T (x) = 1, and [latex]T (x^2) = 2 + 2x[/latex] are written above the columns: [latex]\begin{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ T(1) \ T(x) \ T(x^2)\\ B=\begin{bmatrix} 0&& 1&&2 \\ 0&&0&&2\\0&&0&&0 \end{bmatrix} \end{matrix} \begin{matrix}\\\\1 \\ x \\ x^2 \end{matrix} [/latex]. The last column of B, for example, indicates that [latex]T (x^2)= 2 · 1 + 2 · x = 2 + 2x.[/latex]

Question 4.3.2: Use Theorem 4.3.2 to find the matrix B of the linear transfo......

Linear Algebra with Applications [1016048]

Use Theorem 4.3.2 to find the matrix B of the linear transformation $T ( f ) = f^{\prime} + f^{\prime \prime}$ from $P_2$ to $P_2$ with respect to the standard basis $𝔅 = (1, x, x^2)$ ; see Example 1.

Step-by-Step

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By Theorem 4.3.2, we have

$B=\begin{bmatrix}[T(1)]_𝔅&[T(x)]_𝔅&[T(x^2)]_𝔅\end{bmatrix}$ .

Now

As in Section 3.4, it is suggestive to write the basis elements 1, x, and $x^2$ next to the rows of matrix B, while the values T (1) = 0, T (x) = 1, and $T (x^2) = 2 + 2x$ are written above the columns:

$\begin{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ T(1) \ T(x) \ T(x^2)\\ B=\begin{bmatrix} 0&& 1&&2 \\ 0&&0&&2\\0&&0&&0 \end{bmatrix} \end{matrix} \begin{matrix}\\\\1 \\ x \\ x^2 \end{matrix}$ .

The last column of B, for example, indicates that $T (x^2)= 2 \cdot 1 + 2 \cdot x = 2 + 2x.$