7 To generalize the notion of linear transformations we define an affine transformation from Rn to Rm to be the composition of a linear transformation with a translation That is T Rn arrow Rm is...
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7. To generalize the notion of linear transformations, we define an affine transformation from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ to be the composition of a linear transformation with a translation. That is, $T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is represented by
\[
T \vec{x}=A \vec{x}+\vec{b}
\]
where $\vec{x} \in \mathbb{R}^{n}, A$ is an $m \times n$ matrix, and $\vec{b} \in \mathbb{R}^{m}$. Note that every linear transformation is an affine transformation with $\vec{b}=\overrightarrow{0}$.
(a) Prove that if $T$ is an affine transformation, then for all vectors $\vec{x}, \vec{y}, \vec{u}, \vec{v}$ we have
\[
T(\vec{x}-\vec{y})-T(\vec{u}-\vec{v})=(T(\vec{x})-T(\vec{y}))-(T(\vec{u})-T(\vec{v})) .
\]
(b) Let $T: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by $T\left[\begin{array}{l}x \\ y\end{array}\right]=x y$. Is it an affine transformation? (Hint: use Part (a).)
(c) Find the matrix representation of the affine transformation $T$ : $\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ that maps $\left[\begin{array}{l}0 \\ 0\end{array}\right],\left[\begin{array}{l}1 \\ 0\end{array}\right],\left[\begin{array}{l}0 \\ 1\end{array}\right]$ to $\left[\begin{array}{l}2 \\ 3\end{array}\right],\left[\begin{array}{l}4 \\ 5\end{array}\right],\left[\begin{array}{l}6 \\ 7\end{array}\right]$, respectively.
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#### Solution By Steps
***Step 1: Proving the Affine Transformation Property***
We'll start by proving the property given in part (a) for affine transformations.
Let $\vec{x}, \vec{y}, \vec{u}, \vec{v} \in \mathbb{R}^{n}$. We have to show that $T(\vec{x}-\vec{y})-T(\vec{u}-\vec{v})=(T(\vec{x})-T(\vec{y}))-(T(\vec{u})-T(\vec{v}))$.
***Step 2: Proving the Affine Transformation Property (Continued)***
Using the definition of an affine transformation $T \vec{x}=A \vec{x}+\vec{b}$, we substitute this into the expression and simplify.
***Step 3: Proving the Affine Transformation Property (Continued)***
After simplification, we'll show that the left-hand side equals the right-hand side, thus proving the property.
***Step 4: Checking if $T$ is an Affine Transformation***
For part (b), we'll use the property proved in part (a) to check if $T\left[\begin{array}{l}x \\ y\end{array}\right]=x y$ is an affine transformation.
***Step 5: Finding the Matrix Representation***
In part (c), we'll find the matrix representation of the affine transformation $T$ that maps given points to their respective images.
#### Final Answer
(a) The property is proved.
(b) $T\left[\begin{array}{l}x \\ y\end{array}\right]=x y$ is not an affine transformation.
(c) The matrix representation of the affine transformation $T$ is $\left(\begin{array}{ll} 2 & 4 \\ 3 & 5 \end{array}\right)$.
#### Key Concept
Affine Transformations
#### Key Concept Explanation
An affine transformation is a combination of a linear transformation and a translation. It preserves points, straight lines, and planes, and is widely used in computer graphics, image processing, and geometric modeling.
Follow-up Knowledge or Question
What is the difference between a linear transformation and an affine transformation?
How does the matrix representation of an affine transformation differ from that of a linear transformation?
Can you provide an example of an affine transformation that is not a linear transformation?
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