One-to-one and Onto Transformations

Introduction

One-to-one Transformations

Onto Transformations

Comparison

Let

and let $T(x)=Ax$, so $T\colon\mathbb{R}^2\to\mathbb{R}^3$. Which statement is correct?

A $3\times 5$ matrix $A$ has rank $3$. What can you say about the transformation $T(x)=Ax$?

In one word: a transformation $T\colon\mathbb{R}^n\to\mathbb{R}^m$ is ______ if for every vector $b$ in $\mathbb{R}^m$ the equation $T(x)=b$ has at least one solution.

Let

and let $T(x)=Ax$. Enter the dimension of the range of $T$ (a number).

Let

so that $T(x)=Ax$ maps $\mathbb{R}^5$ to $\mathbb{R}^3$. Enter the dimension of the null space of $A$ (a number).

For the matrix transformation $T(x) = Ax$, the transformation is one-to-one exactly when which condition holds?

A $2\times 3$ matrix $A$ has columns that span $\mathbb{R}^2$. How many free variables does $Ax = 0$ have?