can you solve this question: let B = [1, 1, 1], [-1, 0, -2], [1, 0, 3] let C = [1, 0, 0, 0], [1, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1] let M = [3, -2, 2/ 0, -1, 2/ 6, -4, 4/ -9, 5, -4] be the matrix transformation of T from basis B to C the closed from of the transformation T[1, b, c] = ?

To find the closed form of the transformation \( T \) for any vector \( [1, b, c] \) in terms of the transformation matrix \( M \), we need to analyze how \( M \) acts on vectors in the basis \( B \) and maps them to basis \( C \).

Given dataTo find the closed form of the transformation \( T \) for any vector \( [1, b, c] \) in terms of the transformation matrix \( M \), we need to analyze how \( M \) acts on vectors in the basis \( B \) and maps them to basis \( C \).

Given 

1. **Basis \( B \)**: 
   \[
   B = \left\{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ -2 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} \right\}
   \]
   
2. **Basis \( C \)**:
   \[
   C = \left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}
   \]

3. **Matrix \( M \)**, which represents the transformation \( T \) from basis \( B \) to \( C \):
   \[
   M = \begin{bmatrix} 3 & -2 & 2 \\ 0 & -1 & 2 \\ 6 & -4 & 4 \\ -9 & 5 & -4 \end{bmatrix}
   \]

### Steps to Determine \( T([1, b, c]) \)

1. **Express** \( [1, b, c] \) as a linear combination of the basis \( B \):
   \[
   [1, b, c] = x_1 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + x_2 \begin{bmatrix} -1 \\ 0 \\ -2 \end{bmatrix} + x_3 \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix}
   \]
   This equation allows us to find the coordinates \( [x_1, x_2, x_3] \) in the basis \( B \).

2. **Solve for** \( x_1 \), \( x_2 \), and \( x_3 \) by setting up equations for each component:
   - From the \( x \)-component: \( 1 = x_1 - x_2 + x_3 \)
   - From the \( y \)-component: \( b = x_1 \)
   - From the \( z \)-component: \( c = x_1 - 2x_2 + 3x_3 \)

   Using these equations, we solve for \( x_1 \), \( x_2 \), and \( x_3 \) in terms of \( b \) and \( c \).

3. **Apply the Transformation Matrix \( M \)**:
   Once we have the coordinates \( [x_1, x_2, x_3] \), we multiply \( M \) by this vector to get the image of \( [1, b, c] \) under \( T \) in basis \( C \).

Would you like to go through the detailed algebraic solution for the coefficients \( x_1 \), \( x_2 \), and \( x_3 \), or apply the matrix transformation directly? 

Here are some questions to further explore:
1. How do we determine coordinates in different bases for a given vector?
2. What properties define a matrix transformation in a particular basis?
3. How do we verify if a transformation matrix is correctly mapped between bases?
4. How does a transformation matrix change if the bases are switched?
5. How can we interpret the transformation geometrically?

**Tip:** When working with basis transformations, consistently label your coordinates with respect to each basis to avoid confusion.

can you solve this question: let B = [1, 1, 1], [-1, 0, -2], [1, 0, 3] let C = [1, 0, 0, 0], [1, 1, 1, 1], [0, 0, 1, 1], [0, 0, 0, 1] let M = [3, -2, 2/ 0, -1, 2/ 6, -4, 4/ -9, 5, -4] be the matrix transformation of T from basis B to C the closed form of the transformation T[1, b, c] = ?

Explore how to determine the closed form of a matrix transformation T for a vector in terms of transformation matrix M between basis B and C. This solution covers key concepts in linear algebra, including basis transformations, matrix representation, and coordinate conversions. Suitable for college-level students.

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