📚 Topic Summary
In linear algebra, the composition of linear transformations refers to applying one linear transformation after another. If you have two linear transformations, $T: V \rightarrow W$ and $U: W \rightarrow Z$, their composition, denoted as $U \circ T$, is a new linear transformation that maps vectors from $V$ to $Z$. Essentially, you first apply $T$ to a vector in $V$, which gives you a vector in $W$, and then you apply $U$ to that resulting vector in $W$ to get a vector in $Z$. Understanding composition is crucial for simplifying complex transformations and analyzing their properties.
The matrix representation of $U \circ T$ is obtained by multiplying the matrix representing $U$ by the matrix representing $T$. Order matters: $U \circ T$ is generally different from $T \circ U$! Composition allows us to build complex transformations from simpler ones, offering a powerful tool for problem-solving in linear algebra.
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Linear Transformation |
A. The act of applying one transformation after another. |
| 2. Composition |
B. A mapping between vector spaces that preserves vector addition and scalar multiplication. |
| 3. Vector Space |
C. A set of objects that can be added together and multiplied by scalars. |
| 4. Matrix Representation |
D. A rectangular array of numbers representing a linear transformation. |
| 5. Scalar Multiplication |
E. An operation that multiplies a vector by a scalar, resulting in another vector. |
Matchings: 1-B, 2-A, 3-C, 4-D, 5-E
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
The __________ of two linear transformations, $T$ and $U$, is denoted as $U \circ T$. This means applying $T$ first, followed by __________. The matrix representing the composite transformation $U \circ T$ is found by __________ the matrix of $U$ by the matrix of $T$. The order of transformations __________ because $U \circ T$ is not always the same as $T \circ U$. Therefore, understanding the order is __________ when composing linear transformations.
Answers: composition, $U$, multiplying, matters, essential
🤔 Part C: Critical Thinking
Explain, in your own words, why the order of linear transformations matters when composing them. Provide a simple example to illustrate your point.