Let _ R2 be a line through the origin. a. Give a geometric argument that reflection

Chapter 2, Problem 15

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QUESTION:

Let \(\ell \subset \mathbb{R}^2\) be a line through the origin.

a. Give a geometric argument that reflection across \(\ell\), the function \(R_{\ell}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), is a linear transformation. (Hint: Consider the right triangles formed by \(\mathbf{x}\) and \(\mathbf{x}^{\|}, \mathbf{y}\) and \(\mathbf{y}^{\|}\), and \(\mathbf{x}+\mathbf{y}\) and \(\mathbf{x}^{\|}+\mathbf{y}^{\|}\).)

b. Give a geometric argument that projection onto \(\ell\), the function \(P_{\ell}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), is a linear transformation.

Questions & Answers

QUESTION:

Let \(\ell \subset \mathbb{R}^2\) be a line through the origin.

a. Give a geometric argument that reflection across \(\ell\), the function \(R_{\ell}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), is a linear transformation. (Hint: Consider the right triangles formed by \(\mathbf{x}\) and \(\mathbf{x}^{\|}, \mathbf{y}\) and \(\mathbf{y}^{\|}\), and \(\mathbf{x}+\mathbf{y}\) and \(\mathbf{x}^{\|}+\mathbf{y}^{\|}\).)

b. Give a geometric argument that projection onto \(\ell\), the function \(P_{\ell}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), is a linear transformation.

ANSWER:

Problem 15

Let  be a line through the origin.

     a. Give a geometric argument that the reflection across , the function , is a linear transformation. (Hint: consider the right triangles formed by x and , y and , and  and.)

     b. Give a geometric argument that projection onto , the function , is a linear transformation.

Step by step solution

Step 1 of 3

(a)

Consider the following figure,

Here point C divides the line AB in the ratio .

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