# Image and range of linear transformations

### Image and range of linear transformations

#### Lessons

Recall the matrix equation
$Ax=b$

Normally, we say that the product of $A$ and $x$ gives $b$. Now we are going to say that $A$ is a transformation matrix that transforms a vector $x$ into a vector $b$ (we call $b$ an image of $x$).

In a sense $Ax$ is a function where if we plug in a vector, then it spits out another vector. If we call this function $T(x)$, then
$T(x)=Ax$

where $T$ is the transformation. Note that $T(x)$ is an image $x$ since $T(x)=b$.

We say that a vector $c$ is in a range of the transformation $T$ if there exists a $x$ where:

$T(x)=c$
• 1.
Image and Range of Linear Transformations Overview:
a)
Matrix Transformations
• Transforming from $x$ to $b$
• How transforming vector look like visually

b)
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 $T(x)$

c)
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 $T$

• 2.
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

• 3.
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.

• 4.
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 $T$.

• 5.
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.