To find the change-of-basis matrix $P_{B \leftarrow C}$, which converts coordinates from basis $C$ to basis $B$, we need to express each vector in basis $C$ in terms of the basis $B$.

Given

Basis $B$:

$B = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}$

Basis $C$:

$C = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}$

We need to write the vectors of $C$ in terms of the standard basis $B$.

Step-by-Step Solution

To find the change-of-basis matrix $P_{B \leftarrow C}$, we can use the following method:

1. Construct the matrix $C$ whose columns are the coordinates of the vectors of $C$ in the standard basis $B$.
2. Since the basis $B$ is the standard basis, the change-of-basis matrix $P_{B \leftarrow C}$ is simply the inverse of matrix $C$.

Let's construct matrix $C$: $C = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}$

Next, we find the inverse of matrix $C$ to obtain $P_{B \leftarrow C}$.

Finding the Inverse of $C$

The matrix $C$ is: $C = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}$

To find $C^{-1}$, we perform the following steps:

1. Augment $C$ with the identity matrix:

1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right)$$ 2. Use row operations to reduce $$C$$ to the identity matrix. **Step 1**: Subtract the first row from the second and third rows: $$\left( \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 1 & 0 \\ 0 & 1 & 1 & -1 & 0 & 1 \end{array} \right)$$ **Step 2**: Subtract the second row from the third row: $$\left( \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 1 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 \end{array} \right)$$ Now, we have: $$\left( \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 1 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 \end{array} \right)$$ The augmented part is now the inverse of $$C$$: $$C^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ So, the change-of-basis matrix $$P_{B \leftarrow C}$$ is: $$P_{B \leftarrow C} = \begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ ### Summary The change-of-basis matrix from basis $$C$$ to basis $$B$$ is: $$P_{B \leftarrow C} = \begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix}$$ Would you like details or have any questions? Here are 8 related questions that expand on this information: 1. How do you find the change-of-basis matrix in general? 2. What are the properties of the change-of-basis matrix? 3. How can you verify that a matrix is the inverse of another matrix? 4. How do you apply the change-of-basis matrix to a vector? 5. What is the significance of the change-of-basis matrix in linear transformations? 6. How do you find the coordinates of a vector in a new basis using the change-of-basis matrix? 7. What are some practical applications of change-of-basis matrices? 8. How do you determine if a set of vectors forms a basis for $$\mathbb{R}^n$$? **Tip:** When computing the inverse of a matrix, always double-check your row operations to ensure accuracy.