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 $T[a, b, c]$.