Study Note: Composition of Transformations

When working with linear transformations, you often apply one transformation and then another. For example, imagine you first rotate a plane by 90 degrees counterclockwise and then apply a shear transformation. The result of doing both steps one after the other is a new transformation that combines the effects of both. This new transformation is called the "composition" of the two separate transformations.

In linear algebra, every transformation can be represented by a matrix. When you combine (compose) two transformations, you can also represent the combined result using a matrix.

Example: Rotation and Shear

1. Start with the basis vectors:
• i is the unit vector pointing right, which is initially [1,0].
• j is the unit vector pointing up, which is initially [0,1].
2. Step 1: Rotation:
• After rotating the plane by 90° counterclockwise, i moves to [0,1], and j moves to [−1,0].
3. Step 2: Shear:
• After applying the shear, i ends up at [1,1], and j moves to [−1,0].
4. Result: The new transformation (after both the rotation and shear) is captured by a matrix where:
• The first column is [1,1] (the final position of i^).
• The second column is [−1,0] (the final position of j^).

This combined matrix gives a single transformation that describes the effect of rotating the plane and then applying a shear, all in one step. So, instead of thinking of the two actions separately, you can think of them as one combined transformation represented by the matrix.

Composition is Multiplication

MATRIX General Form

Non-commutativity

Aside the above general rules that the matrix does not commute, indentity matrix is an exception to that rule. The detail of the exception is given below:

PRActical Examples