The composition of linear transformations is the application of one linear transformation followed by another.
In mathematics, particularly in linear algebra, a linear transformation (also known as a linear map or linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
When we talk about the composition of linear transformations, we are essentially describing the process of performing one linear transformation and then applying another linear transformation to the result of the first one.
Let's say you have two linear transformations:
- $T_1: V \rightarrow W$ which transforms vectors from vector space $V$ to vector space $W$.
- $T_2: W \rightarrow U$ which transforms vectors from vector space $W$ to vector space $U$.
The composition of $T_2$ with $T_1$, often denoted as $T_2 \circ T_1$, is a new transformation that maps vectors directly from $V$ to $U$. For any vector $\mathbf{v}$ in $V$, the composite transformation works as follows:
$(T_2 \circ T_1)(\mathbf{v}) = T_2(T_1(\mathbf{v}))$
This means you first apply $T_1$ to $\mathbf{v}$ to get a vector in $W$, and then apply $T_2$ to that resulting vector in $W$ to get a final vector in $U$.
Key Properties of Composite Linear Transformations
As highlighted in the reference, the composition of two or more linear maps retains the fundamental property of linearity:
- Linearity Preservation: The composite transformation $T_2 \circ T_1$ is also a linear transformation. This means it will also preserve vector addition and scalar multiplication, mapping the zero vector to the zero vector and preserving the structure of the vector spaces.
The Matrix of the Composite Transformation
A crucial aspect of linear transformations is their representation using matrices. If $T_1$ and $T_2$ are linear transformations, they can be represented by matrices.
The reference explicitly states: "the matrix of the composite transformation is equal to the product of the matrices of the two original maps."
Let's denote the matrix representing $T_1$ as $M_1$ and the matrix representing $T_2$ as $M_2$.
For the composition $T_2 \circ T1$, the matrix representing this composite transformation, $M{T_2 \circ T_1}$, is given by the matrix product $M_2 M_1$.
It is important to note that the order of matrix multiplication matters. The matrix of the second transformation ($M_2$) is multiplied by the matrix of the first transformation ($M_1$). This corresponds to the order in which the transformations are applied to a vector: $T_2(T_1(\mathbf{v}))$ corresponds to $M_2 (M_1 \mathbf{v})$.
Practical Example (Conceptual)
Imagine two transformations in a 2D plane:
- $T_1$: Rotation by 45 degrees.
- $T_2$: Scaling by a factor of 2.
The composition $T_2 \circ T_1$ would first rotate a vector by 45 degrees and then scale the rotated vector by 2. This composite transformation ($T_2 \circ T_1$) is itself a linear transformation. Its matrix would be the product of the matrix for scaling ($M_2$) and the matrix for rotation ($M_1$).
The composition $T_1 \circ T_2$ would first scale a vector by 2 and then rotate the scaled vector by 45 degrees. This is a different composite transformation, and its matrix would be the product of the matrix for rotation ($M_1$) and the matrix for scaling ($M_2$), i.e., $M_1 M_2$. Since matrix multiplication is generally not commutative, $M_1 M_2 \neq M_2 M_1$, illustrating that the order of composition matters.
Summary Table
| Aspect | Description | Property for Composition $(T_2 \circ T_1)$ |
|---|---|---|
| Definition | Applying one transformation after another. | $(T_2 \circ T_1)(\mathbf{v}) = T_2(T_1(\mathbf{v}))$ |
| Input/Output | $T_1: V \to W$, $T_2: W \to U$ | $(T_2 \circ T_1): V \to U$ |
| Linearity | Preserves vector addition and scalar multiplication. | The composite transformation is also linear. |
| Matrix Form | Represented by matrices $M_1$ and $M_2$. | Matrix of composite is $M_2 M_1$ (product of original matrices) |
Composition of linear transformations is a fundamental concept in linear algebra, allowing us to combine transformations and represent complex sequences of operations with a single composite transformation and its corresponding matrix product.
