Questions: Consider the ordered bases B=1+2x, 2+5x and C=3x-1,3+3x for the vector space P2. a. Find the transition matrix from C to the standard ordered basis E=1, x. b. Find the transition matrix from B to E. c. Find the transition matrix from E to B. d. Find the transition matrix from C to B. e. Find the coordinates of p(x)=2x-2 in the ordered basis B. f. Find the coordinates of q(x) in the ordered basis B if the coordinate vector of q(x) in C is [q(x)]C=[2 -2].

Consider the ordered bases B=1+2x, 2+5x and C=3x-1,3+3x for the vector space P2.
a. Find the transition matrix from C to the standard ordered basis E=1, x.
b. Find the transition matrix from B to E.
c. Find the transition matrix from E to B.
d. Find the transition matrix from C to B.
e. Find the coordinates of p(x)=2x-2 in the ordered basis B.
f. Find the coordinates of q(x) in the ordered basis B if the coordinate vector of q(x) in C is [q(x)]C=[2 -2].

Transcript text: Consider the ordered bases $\mathcal{B}=\{1+2 x, 2+5 x\}$ and $\mathcal{C}=\{3 x-1,3+3 x\}$ for the vector space $\mathbb{P}_{2}$. a. Find the transition matrix from $\mathcal{C}$ to the standard ordered basis $\mathcal{E}=\{1, x\}$. b. Find the transition matrix from $\mathcal{B}$ to $\mathcal{E}$. c. Find the transition matrix from $\mathcal{E}$ to $\mathcal{B}$. d. Find the transition matrix from $\mathcal{C}$ to $\mathcal{B}$. e. Find the coordinates of $p(x)=2 x-2$ in the ordered basis $\mathcal{B}$. f. Find the coordinates of $q(x)$ in the ordered basis $\mathcal{B}$ if the coordinate vector of $q(x)$ in $\mathcal{C}$ is $[q(x)]_{\mathcal{C}}=\left[\begin{array}{c}2 \\ -2\end{array}\right]$.

Solution

Solution Steps

To solve the given problems, we need to find transition matrices between different bases for the vector space $\mathbb{P}_{2}$. The transition matrix from one basis to another can be found by expressing each vector of the first basis in terms of the second basis. For the coordinates of a polynomial in a given basis, we express the polynomial as a linear combination of the basis vectors.

a. To find the transition matrix from $\mathcal{C}$ to the standard basis $\mathcal{E}$, express each vector in $\mathcal{C}$ in terms of $\mathcal{E}$.

b. Similarly, express each vector in $\mathcal{B}$ in terms of $\mathcal{E}$ to find the transition matrix from $\mathcal{B}$ to $\mathcal{E}$.

c. The transition matrix from $\mathcal{E}$ to $\mathcal{B}$ is the inverse of the matrix found in part b.

Step 1: Transition Matrix from $\mathcal{C}$ to $\mathcal{E}$

To find the transition matrix from the basis $\mathcal{C} = \{3x - 1, 3 + 3x\}$ to the standard basis $\mathcal{E} = \{1, x\}$, we express each vector in $\mathcal{C}$ in terms of $\mathcal{E}$. The matrix $\mathcal{C}$ is given by:

\[ C = \begin{bmatrix} -1 & 3 \\ 3 & 3 \end{bmatrix} \]

The transition matrix $T_{\mathcal{C}}^{\mathcal{E}}$ is the inverse of $C$:

\[ T_{\mathcal{C}}^{\mathcal{E}} = \begin{bmatrix} -0.25 & 0.25 \\ 0.25 & 0.0833 \end{bmatrix} \]

Step 2: Transition Matrix from $\mathcal{B}$ to $\mathcal{E}$

For the basis $\mathcal{B} = \{1 + 2x, 2 + 5x\}$, we express each vector in $\mathcal{B}$ in terms of $\mathcal{E}$. The matrix $\mathcal{B}$ is given by:

\[ B = \begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} \]

The transition matrix $T_{\mathcal{B}}^{\mathcal{E}}$ is the inverse of $B$:

\[ T_{\mathcal{B}}^{\mathcal{E}} = \begin{bmatrix} 5 & -2 \\ -2 & 1 \end{bmatrix} \]

Step 3: Transition Matrix from $\mathcal{E}$ to $\mathcal{B}$

The transition matrix from $\mathcal{E}$ to $\mathcal{B}$ is the inverse of $T_{\mathcal{B}}^{\mathcal{E}}$:

\[ T_{\mathcal{E}}^{\mathcal{B}} = \begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} \]