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Let _ R2 be a line through the origin. a. Give a geometric argument that reflection
Chapter 2, Problem 15(choose chapter or problem)
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 .