# Image and range of linear transformations

##### Intros
###### Lessons
1. Image and Range of Linear Transformations Overview:
2. Matrix Transformations
• Transforming from $x$ to $b$
• How transforming vector look like visually
3. The Image of $x$
$T(x)$: the image of $x$ under the transformation $T$
• Finding the image $T(x)$ when given $x$
• Finding $x$ when given the image $T(x)$
4. The Range of $T$
• The set of all images $T(x)$
• What the range looks like visually
• How to know if a vector is in the range of $T$
##### Examples
###### Lessons
1. Consider the matrix , and let's define $T: \Bbb{R}^4$$\Bbb{R}^3$ by $T(x)=AX$. Find the images under $T$ of and
1. Finding $x$ when given the image under $T$
Let's define $T: \Bbb{R}^3$$\Bbb{R}^2$ by $T(x)=Ax$. Let

Find the vector $x$ whose image under $T$ is $b$, and find out whether $x$ is unique.
1. A vector in the Range of $T$
Let's define $T: \Bbb{R}^2$$\Bbb{R}^3$ by $T(x)=Ax$. Let

Determine if $b$ is in the range of the transformation $T$.
1. Geometric Interpretation of $T$
Use a graph to plot the vector and its image under the transformation T. You are given that:

Explain what the transformation did to the vector.