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.