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[a, b, c] = ?

To solve for the transformation \( T[a, b, c] \) in closed form, let's analyze the components given:

1. **Basis \( B \):** 
   \[
   B = \begin{bmatrix} 1 & 1 & 1 \\ -1 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}
   \]
   This represents a basis for \( \mathbb{R}^3 \).

2. **Basis \( C \):** 
   \[
   C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix}
   \]
   This is a basis for \( \mathbb{R}^4 \), where each column represents one of the basis vectors in \( C \).

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

We are tasked with finding the closed form of the transformation \( T[a, b, c] \) in the standard basis. 

### Step 1: Interpretation of \( M \)
The matrix \( M \) describes how vectors in basis \( B \) (for the domain) map into vectors in basis \( C \) (for the codomain) under the transformation \( T \). To find \( T \) in standard coordinates, we need to:

1. Express the standard basis in terms of \( B \).
2. Express the columns of \( M \) (which are \( T \)-images of the basis \( B \)-vectors) in terms of the standard coordinates using the basis \( C \).

### Step 2: Converting Basis \( B \) to Standard Basis
We can write any vector \( [a, b, c]^T \) in the standard basis in terms of the basis \( B \):
\[
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = B \begin{bmatrix} a \\ b \\ c \end{bmatrix}.
\]
This gives us the coordinates of \( [a, b, c]^T \) in terms of basis \( B \).

### Step 3: Applying Matrix \( M \) and Converting Back to Standard Basis
To apply the transformation \( T \) in standard coordinates:
1. Multiply \( [a, b, c]^T \) by \( M \) to get the coordinates in basis \( C \).
2. Map those coordinates back to the standard basis by using the matrix \( C \).

This process would yield the closed-form transformation \( T[a, b, c] \).

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[a, b, c] = ?

Explore how to find the closed-form expression for the matrix transformation T[a, b, c] from basis B to basis C, with matrix M representing the transformation. This problem involves key linear algebra concepts like matrix transformations and change of basis. Suitable for undergraduate students.

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