Question 1165864
This problem asks you to confirm that a given function $T: \mathbb{R}^3 \to \mathbb{R}^2$ is a **linear transformation** and then find its **standard matrix** $A$.

The given transformation is:
$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + x_3 \end{pmatrix}$$

---

## 1. Showing $T$ is a Linear Transformation

A function $T$ is a linear transformation if it satisfies two properties for any vectors $\mathbf{u}$ and $\mathbf{v}$ in the domain ($\mathbb{R}^3$) and any scalar $c$ in $\mathbb{R}$:

1.  **Additivity:** $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2.  **Homogeneity:** $T(c\mathbf{u}) = cT(\mathbf{u})$

Let $\mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}$.

### 1.1. Additivity

$$\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix}$$

$$\begin{aligned} T(\mathbf{u} + \mathbf{v}) &= T \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ u_3 + v_3 \end{pmatrix} \\ &= \begin{pmatrix} (u_1+v_1) + 2(u_2+v_2) + 3(u_3+v_3) \\ -3(u_1+v_1) + 2(u_2+v_2) + (u_3+v_3) \end{pmatrix} \\ &= \begin{pmatrix} (u_1 + 2u_2 + 3u_3) + (v_1 + 2v_2 + 3v_3) \\ (-3u_1 + 2u_2 + u_3) + (-3v_1 + 2v_2 + v_3) \end{pmatrix} \\ &= \begin{pmatrix} u_1 + 2u_2 + 3u_3 \\ -3u_1 + 2u_2 + u_3 \end{pmatrix} + \begin{pmatrix} v_1 + 2v_2 + 3v_3 \\ -3v_1 + 2v_2 + v_3 \end{pmatrix} \\ &= T(\mathbf{u}) + T(\mathbf{v}) \end{aligned}$$
**Additivity is satisfied.**

### 1.2. Homogeneity

$$c\mathbf{u} = \begin{pmatrix} cu_1 \\ cu_2 \\ cu_3 \end{pmatrix}$$

$$\begin{aligned} T(c\mathbf{u}) &= T \begin{pmatrix} cu_1 \\ cu_2 \\ cu_3 \end{pmatrix} \\ &= \begin{pmatrix} (cu_1) + 2(cu_2) + 3(cu_3) \\ -3(cu_1) + 2(cu_2) + (cu_3) \end{pmatrix} \\ &= \begin{pmatrix} c(u_1 + 2u_2 + 3u_3) \\ c(-3u_1 + 2u_2 + u_3) \end{pmatrix} \\ &= c \begin{pmatrix} u_1 + 2u_2 + 3u_3 \\ -3u_1 + 2u_2 + u_3 \end{pmatrix} \\ &= cT(\mathbf{u}) \end{aligned}$$
**Homogeneity is satisfied.**

Since both properties hold, $T$ is a **linear transformation**. (In general, any transformation defined by a system of linear equations with no constant terms is a linear transformation.)

---

## 2. Determining the Standard Matrix $A$

The standard matrix $A$ of a linear transformation $T$ is the matrix whose columns are the images of the **standard basis vectors** of the domain ($\mathbb{R}^3$):
$$A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & T(\mathbf{e}_3) \end{bmatrix}$$
where $\mathbf{e}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, and $\mathbf{e}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$.

1.  **Image of $\mathbf{e}_1$:**
    $$T \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} (1) + 2(0) + 3(0) \\ -3(1) + 2(0) + (0) \end{pmatrix} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}$$

2.  **Image of $\mathbf{e}_2$:**
    $$T \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} (0) + 2(1) + 3(0) \\ -3(0) + 2(1) + (0) \end{pmatrix} = \begin{pmatrix} 2 \\ 2 \end{pmatrix}$$

3.  **Image of $\mathbf{e}_3$:**
    $$T \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} (0) + 2(0) + 3(1) \\ -3(0) + 2(0) + (1) \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$$

### The Matrix $A$

Assembling the column vectors:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix}$$

We can verify that $T(\mathbf{x}) = A\mathbf{x}$:
$$A\mathbf{x} = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 1x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + 1x_3 \end{pmatrix} = T(\mathbf{x})$$

The standard matrix is:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix}$$