Find the change-of-basis matrix 

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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:
\[ \left( C | I \right) = \left( \begin{array}{ccc|ccc}
1 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 & 0 & 1
\end{array} \right) \]

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.

Learn how to find the change-of-basis matrix P from basis C to basis B in linear algebra. This detailed solution covers constructing matrices, finding inverses, and applying change-of-basis matrices. Suitable for advanced undergraduate students and those studying linear algebra.

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