To find the matrix representation $A$ of the linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^2$, we assume $T(\mathbf{x}) = A \mathbf{x}$ for $\mathbf{x} \in \mathbb{R}^3$. Matrix $A$ will be a $2 \times 3$ matrix since $T$ maps from $\mathbb{R}^3$ to $\mathbb{R}^2$.

The problem provides the following information: $T\left(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\right) = \begin{bmatrix} 0 \\ -2 \end{bmatrix}, \quad T\left(\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -2 \end{bmatrix}, \quad T\left(\begin{bmatrix} 1 \\ 4 \\ 9 \end{bmatrix}\right) = \begin{bmatrix} 3 \\ 0 \end{bmatrix}.$

Let the matrix $A$ be: $A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}$

We are given three equations based on the images of the vectors under $T$:

1. $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -2 \end{bmatrix}$
2. $A \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 1 \\ -2 \end{bmatrix}$
3. $A \begin{bmatrix} 1 \\ 4 \\ 9 \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \end{bmatrix}$

Using these equations, we can set up the following system of linear equations for each row of $A$:

First row:

1. $a + b + c = 0$
2. $a + 2b + 3c = 1$
3. $a + 4b + 9c = 3$

Second row:

1. $d + e + f = -2$
2. $d + 2e + 3f = -2$
3. $d + 4e + 9f = 0$

Let's solve these systems separately to determine the values of $a, b, c$ and $d, e, f$.

Solving for $a, b, c$:

Using the equations:

1. $a + b + c = 0$
2. $a + 2b + 3c = 1$
3. $a + 4b + 9c = 3$

Subtract the first equation from the second: $b + 2c = 1$ Subtract the second equation from the third: $2b + 6c = 2 \Rightarrow b + 3c = 1$

Now we have the system:

1. $b + 2c = 1$
2. $b + 3c = 1$

Subtract the first from the second: $c = 0$ Substitute $c = 0$ into $b + 2c = 1$: $b = 1$ Finally, substitute $b = 1$ and $c = 0$ into $a + b + c = 0$: $a + 1 + 0 = 0 \Rightarrow a = -1$

So, $a = -1$, $b = 1$, $c = 0$.

Solving for $d, e, f$:

Using the equations:

1. $d + e + f = -2$
2. $d + 2e + 3f = -2$
3. $d + 4e + 9f = 0$

Subtract the first equation from the second: $e + 2f = 0$ Subtract the second equation from the third: $2e + 6f = 2 \Rightarrow e + 3f = 1$

Now we have the system:

1. $e + 2f = 0$
2. $e + 3f = 1$

Subtract the first from the second: $f = 1$ Substitute $f = 1$ into $e + 2f = 0$: $e + 2 = 0 \Rightarrow e = -2$ Finally, substitute $e = -2$ and $f = 1$ into $d + e + f = -2$: $d - 2 + 1 = -2 \Rightarrow d = -1$

So, $d = -1$, $e = -2$, $f = 1$.

Matrix $A$

The matrix representation of $T$ is: $A = \begin{bmatrix} -1 & 1 & 0 \\ -1 & -2 & 1 \end{bmatrix}$

Would you like further details on the steps, or do you have any questions?

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Here are 5 related questions:

1. How does one verify that $A$ is indeed the correct matrix for $T$?
2. What properties of linear transformations does this example illustrate?
3. How would the process change if $T$ were a transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ instead?
4. How can one determine if a linear transformation is invertible from its matrix?
5. What is the geometric interpretation of this transformation in $\mathbb{R}^2$?

Tip: Always check your solutions by plugging values back into the original equations to ensure accuracy.