Matrix Transformations: Scaling, Rotation, and Shear

1. The Universal Recipe for Finding a Transformation Matrix

The previous lesson hinted at the core idea, and it's so fundamental that we will start by focusing on it exclusively. A 2D linear transformation is completely determined by where it moves the two standard basis vectors: $\hat{\mathbf{i}}$ (the vector $[1,0]$) and $\hat{\mathbf{j}}$ (the vector $[0,1]$).

Why? Because any vector $\mathbf{v}=\left(\begin{array}{c}x\\ y\end{array}\right)$ can be written as a linear combination of these basis vectors: $\mathbf{v}=x\hat{\mathbf{i}}+y\hat{\mathbf{j}}$. Because the transformation $T$ is linear, the transformed vector $T(\mathbf{v})$ will be the same linear combination of the transformed basis vectors:

This gives us a universal recipe:

1. Find where the basis vector $\hat{\mathbf{i}}$ lands after the transformation. These coordinates will be the first column of your matrix.
2. Find where the basis vector $\hat{\mathbf{j}}$ lands. These coordinates will be the second column of your matrix.

That's it. The resulting 2x2 matrix perfectly encodes the transformation.

To see this principle in action, let's watch a segment from the video we saw last time, but with a new focus.

2. Formalizing the Recipe: The Standard Matrix

The intuitive recipe you just saw has a formal name in linear algebra. Let's define the terms properly, which will be helpful for more advanced topics. This aligns with your preference for understanding the theory first.

The key theorem from this reading confirms our recipe: For a linear transformation $T$, its standard matrix $A$ is given by:

where ${\mathbf{e}}_1=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and ${\mathbf{e}}_2=\left(\begin{array}{c}0\\ 1\end{array}\right)$.

3. A Gallery of Geometric Transformations

Now, let's apply this recipe to build the standard matrices for the three key transformations mentioned in our learning outcome: scaling, rotation, and shear.

Scaling

A scaling transformation stretches or compresses space along the axes. Let's say we want to scale by a factor of $k_x$ along the x-axis and $k_y$ along the y-axis.

1. Transform ${\mathbf{e}}_1$: The vector $\left(\begin{array}{c}1\\ 0\end{array}\right)$ is scaled by $k_x$ along the x-axis. It becomes $\left(\begin{array}{c}{k}_x\\ 0\end{array}\right)$. This is our first column.
2. Transform ${\mathbf{e}}_2$: The vector $\left(\begin{array}{c}0\\ 1\end{array}\right)$ is scaled by $k_y$ along the y-axis. It becomes $\left(\begin{array}{c}0\\ k_y\end{array}\right)$. This is our second column.

Combining these gives the general scaling matrix:

A scaling that is the same in all directions ($k_x=k_y=k$) is called a dilation. A scaling of $k=1$ gives the identity matrix, which does nothing.

For a worked-out numerical example, watch this short video segment.

Rotation

Let's find the matrix for a counter-clockwise rotation by an angle $\theta$ about the origin. We need a little trigonometry.

1. Transform ${\mathbf{e}}_1$: The vector $\left(\begin{array}{c}1\\ 0\end{array}\right)$, a point on the positive x-axis, rotates to the point $\left(\begin{array}{c}\cos \theta \\ \sin \theta \end{array}\right)$. This is our first column.
2. Transform ${\mathbf{e}}_2$: The vector $\left(\begin{array}{c}0\\ 1\end{array}\right)$, a point on the positive y-axis, rotates to the point $\left(\begin{array}{c}\cos (\theta +90^{\circ })\\ \sin (\theta +90^{\circ })\end{array}\right)$, which simplifies to $\left(\begin{array}{c}-\sin \theta \\ \cos \theta \end{array}\right)$. This is our second column.

Combining these gives the rotation matrix:

To see this laid out, you can refer to the following resource.

Quick Practice: What is the matrix for a counter-clockwise rotation of 90°?
(Hint: $\cos (90^{\circ })=0$, $\sin (90^{\circ })=1$)
The matrix is $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. This matches the result from the 3Blue1Brown video.

Shear

A shear transformation slants space, keeping one axis fixed.

• Horizontal Shear: The x-axis is fixed, and points are shifted horizontally depending on their y-coordinate. Let $k$ be the shear factor.
  1. Transform ${\mathbf{e}}_1$: The vector $\left(\begin{array}{c}1\\ 0\end{array}\right)$ is on the x-axis, so it doesn't move. It stays at $\left(\begin{array}{c}1\\ 0\end{array}\right)$. First column.
  2. Transform ${\mathbf{e}}_2$: The vector $\left(\begin{array}{c}0\\ 1\end{array}\right)$ is shifted horizontally by $k\times 1$ (since its y-value is 1). It lands at $\left(\begin{array}{c}k\\ 1\end{array}\right)$. Second column.

The horizontal shear matrix is:

• Vertical Shear: Similarly, a vertical shear fixes the y-axis and shifts points vertically based on their x-coordinate. The matrix is:

Let's watch a final clip to see both types of shear transformations calculated numerically.

Conclusion

Today you've reversed the flow of logic from our last lesson and gained a powerful new skill. Instead of just analyzing a given transformation, you can now build one from scratch to achieve a desired geometric outcome.

Key Takeaways:

• The Universal Recipe: Any 2D linear transformation is fully described by a 2x2 matrix whose columns are the coordinates of the transformed basis vectors, $T({\mathbf{e}}_1)$ and $T({\mathbf{e}}_2)$.
• Scaling Matrix: Stretches or compresses along the axes. It's a diagonal matrix: $\left(\begin{array}{cc}{k}_x& 0\\ 0& k_y\end{array}\right)$.
• Rotation Matrix: Rotates the plane counter-clockwise by angle $\theta$. It involves sine and cosine: $\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$.
• Shear Matrix: Slants the plane. A horizontal shear is $\left(\begin{array}{cc}1& k\\ 0& 1\end{array}\right)$, and a vertical shear is $\left(\begin{array}{cc}1& 0\\ k& 1\end{array}\right)$.

Preview of the Next Lesson:
We now know how to represent and build transformations. But what if we want to quantify their effect or reverse them? In the next lesson, we will learn how to compute the determinant and inverse of a matrix. The determinant will give us a single number describing how much a transformation expands or contracts area, and the inverse will give us a new matrix that represents the "undo" operation for the original transformation.