# Image and range of linear transformations

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##### Intros

###### Lessons

**Image and Range of Linear Transformations Overview:**__Matrix Transformations__

• Transforming from $x$ to $b$

• How transforming vector look like visually__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 $T(x)$__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

- 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
**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.**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 $T$.**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.