Question 1165864: Kuttler, pg. 279, Ex. 5.2.8) Consider the following functions T : R3→ R2 . Show that each is a linear transformation and determine for each the matrix A such that
T (x) = A x.
T . (matrix3x1(x1, x2, x3) )= Matrix2x1(x1+2x2+3x3, 2x2−3x1+x3)
Answer by CPhill(2103)   (Show Source):
You can put this solution on YOUR website! 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:
$$T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 + 2x_2 + 3x_3 \\ -3x_1 + 2x_2 + x_3 \end{pmatrix}$$
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## 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.)
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## 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:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ -3 & 2 & 1 \end{pmatrix}$$
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