Explain why the translation function T: ℝ^2 →ℝ^2, defined by T( x y )=( x+a

Name: Problem 4, Chapter 7: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-09-07T00:39:02-08:00
Duration: 1 min 2 s
Channel: Monica Miller
Description: Explain why the translation function T: ℝ^2 →ℝ^2, defined by T( x y )=( x+a y+b ) for a, b ∈ℝ, is almost never linear. Precisely when is it linear? a, b ∈ℝ, is almost never linear. Precisely when is it linear?

step 1 Okay, so it says to what linear function of X does the linear equation AX plus BY equals C corre

Question

Explain why the translation function $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $T\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+a \\ y+b\end{array}\right)$ for $\quad a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear? $a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear?

   Explain why the translation function $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $T\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+a \\ y+b\end{array}\right)$ for $\quad a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear? $a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 7, Problem 4

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Solved by Expert Monica Miller

09/07/2021

Step 1

A function $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ is linear if it satisfies two conditions: Show more…

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Explain why the translation function $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, defined by $T\left(\begin{array}{l}x \\ y\end{array}\right)=\left(\begin{array}{l}x+a \\ y+b\end{array}\right)$ for $\quad a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear? $a, b \in \mathbb{R}$, is almost never linear. Precisely when is it linear?

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Key Concepts

Linear Transformation

A linear transformation is a mapping between vector spaces that preserves vector addition and scalar multiplication. One of the key properties of any linear transformation is that it sends the zero vector to the zero vector, ensuring that the structure of the vector space is maintained.

Translation (as an Affine Transformation)

A translation shifts every vector in a space by a fixed amount. While it preserves geometric properties like parallelism and the ratios of distances (making it an affine transformation), it generally does not send the zero vector to the zero vector unless the shift is zero. This failure to satisfy the property of mapping zero to zero is why a translation is almost never linear, except when the shift vector is the zero vector, in which case it reduces to the identity map.

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