线性变换

Introduction

Linear Transformations: Definition

The Standard Coordinate Vectors

The Matrix of a Linear Transformation

Which of the following transformations from $\mathbb{R}^2$ to $\mathbb{R}^2$ is linear?

Define $T\colon\mathbb{R}^2\to\mathbb{R}^2$ by $T(x,y)=(xy,\ y)$. Which reasoning correctly shows that $T$ is not linear?

For the rest of the book, $e_1,e_2,\ldots,e_n$ denote the ______ coordinate vectors in $\mathbb{R}^n$. Enter one word.

Define $T\colon\mathbb{R}^2\to\mathbb{R}^2$ by $T(x,y)=(3x-2y,\ 5y)$. Enter the entry in the first row and second column of the standard matrix of $T$ (a number).

Define $T\colon\mathbb{R}^4\to\mathbb{R}^5$ by

the transformation that turns the displacements of four masses into the extensions of the five springs holding them. Enter the second column of the standard matrix of $T$ as five numbers separated by commas (for example: 1,0,0,0,0).

A transformation $T\colon \mathbb{R}^n \to \mathbb{R}^m$ is linear when it satisfies which requirements?

A linear $T\colon \mathbb{R}^2 \to \mathbb{R}^2$ satisfies $T(e_1) = (2,1)$ and $T(e_2) = (0,-1)$. Compute $T(3,2)$ and enter the result as $x,y$.