# The matrix of a linear transformation

### The matrix of a linear transformation

#### Lessons

The standard basis, $e_1$ and $e_2$, are unit vectors in $\Bbb{R}^2$ such that:
If are the transformed vectors, then the standard matrix is

Why does that work? Watch the intro video

The standard basis, $e_1$, $e_2$, and $e_3$ are unit vectors in $\Bbb{R}^3$ such that:

If are the transformed vectors, then the standard matrix is

To find the standard basis in $\Bbb{R}^2$ geometrically in a graph we:
1. Draw the standard basis $e_1$ and $e_2$ in the $x_1 x_2$ plane
2. Draw the transformed vectors using the information given
3. Identify the transformed vectors $T(e_1)$, and $T(e_2)$.
4. Combine them to get the standard matrix

Here are the many types of transformations you may see in this section:

Reflections

Vertical/Horizontal Expansions and contractions

Vertical/Horizontal Shears

Projections

Circle Rotation
• 1.
The Matrix of a Linear Transformation Overview:
a)
The Standard Basis and Matrix
$T(x)=Ax, A$: The Standard Matrix
$\Bbb{R}^2$ standard Basis: and
$\Bbb{R}^3$ standard Basis:
• Transformed standard basis
• Finding the Standard Matrix

b)
Finding the Standard Matrix Geometrically in $\Bbb{R}^2$
• Drawing the Standard basis on a Graph
• Identifying the Transformed Standard Basis
• Combining the Standard Basis’ into a Matrix

c)
Types of Geometric Linear Transformations in $\Bbb{R}^2$
• Reflections on a Line
• Horizontal Expansion/Contraction
• Vertical Expansion/Contraction
• Horizontal & Vertical Shears
• Projections
• Rotation Transformations

• 2.
Finding the Standard Matrix with Transformed Basis
Assume that $T$ is a linear transformation. Find the standard matrix of $T$ if where and .

• 3.
Assume that $T$ is a linear transformation. Find the standard matrix of $T$ if where and .

• 4.
Finding the Standard Matrix Geometrically
Assume that $T$ is a linear transformation. Find the standard matrix of $T$ if $T: \Bbb{R}^2$$\Bbb{R}^2$ is a vertical shear transformation that maps $e_1$ to $e_1+2e_2$ and leaves $e_2$ unchanged.

• 5.
Assume that $T$ is a linear transformation. Find the standard matrix of $T$ if $T: \Bbb{R}^2$$\Bbb{R}^2$ first performs a $x_2=x_1$ reflection, and then a $x_1$ reflection.

• 6.
Finding the Matrix Algebraically
Show that $T$ is a linear transformation by finding a matrix $A$ that implements the mapping:

• 7.
Finding for x given the Transformation
Let $T: \Bbb{R}^2$$\Bbb{R}^2$ be a linear transformation such that . Find a $x$ such that