Linear Transformations and Operators
Definition of linear transformations and operators
Matrix of linear transfomration
Composition of linear transformations
Since linear transformations are functions themselves, we can study their composition. If \( L:\mathbb R^m\to\mathbb R^n \) and \( K:\mathbb R^n\to\mathbb R^p \) are two linear transformations than \( K\circ L:\mathbb R^m\to\mathbb R^p \) is a function. Our next result shows that \( K\circ L \) is a linear transformation.
The following theorem states that the linear combination of two linear transformations with the same domain and codomain is linear.
Matrix of the composition. Product of matrices
The matrix \( \hat Q \) is called the product of matrices \( \hat K \) and \( \hat L \) and is denoted as \( \hat Q=\hat K\cdot \hat L \). We say that the matrix \( \hat L \) is of the format \( n\times m \), the matrix \( \hat K \) is of the format \( p\times n \), and the matrix \( \hat Q \) is of the format \( p\times m \).
In the future we will often use interchangeably trnasformations and their matrices and we will use the same letter to denote them. We will also write \( Lu \) instead of \( L(u) \) when we are dealing with a transformation \( L \) and a vector \( u \). This is consistent with the matrix interpretation in which \( L \) is an \( n\times m \) matrix and \( u \) is an \( 1\times m \) matrix.
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