Linear Transformations and Operators

Definition of linear transformations and operators

Definition 1 (linear transformation and linear operator)
 

A function \( L:\mathbb R^k\to\mathbb R^m \) is called a linear transformation if \( L(\alpha u+\beta v)=\alpha L(u)+\beta L(v) \) for all vectors \( u,v\in\mathbb R^k \) and all scalars \( \alpha,\beta\in\mathbb R \). If \( k=m \), the linear transformation is also called linear operator.

Example 1
 
Let \( L:\mathbb R^2\to\mathbb R^2 \) be a function defined in the following way: \begin{eqnarray*}L\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c}x+y\\x-y\end{array}\right].\end{eqnarray*} Prove that \( L \) is a linear operator.

Example 2
 
Assume that \( L:\mathbb R^2\to\mathbb R^2 \) is the linear operator that satisfies \(\displaystyle L\left(\left[\begin{array}{c}1\\0\end{array}\right]\right)=\left[\begin{array}{c}3\\-2\end{array}\right]\) and \(\displaystyle L\left(\left[\begin{array}{c}0\\1\end{array}\right]\right)=\left[\begin{array}{c}2\\5\end{array}\right]\). Determine \(\displaystyle L\left(\left[\begin{array}{c}3\\7\end{array}\right]\right)\).

Example 3
 
Assume that \( L:\mathbb R^2\to\mathbb R^2 \) is the linear operator that satisfies \(\displaystyle L\left(\left[\begin{array}{c}2\\-3\end{array}\right]\right)=\left[\begin{array}{c}1\\-2\end{array}\right]\) and \(\displaystyle L\left(\left[\begin{array}{c}3\\-1\end{array}\right]\right)=\left[\begin{array}{c}2\\1\end{array}\right]\). Determine \(\displaystyle L\left(\left[\begin{array}{c}1\\2\end{array}\right]\right)\).

Matrix of linear transfomration

Theorem 1
 

Assume that \( e_1 \), \( \dots \), \( e_m \) is a basis of \( \mathbb R^m \), and assume that \( f_1 \), \( \dots \), \( f_m \) are vectors from \( \mathbb R^n \). There exists a unique linear transformation \( L:\mathbb R^m\to\mathbb R^n \) such that \( L(e_1)=f_1 \), \( \dots \), \( L(e_m)=f_m \).

Definition 2 (Matrix of linear transformation)
 

Assume that \( L:\mathbb R^m\to\mathbb R^n \) is as linear transformation, and assume that \(\displaystyle e_1=\left[\begin{array}{c} 1\\0\\ \vdots\\ 0\end{array}\right]\), \( \dots \), \(\displaystyle e_m=\left[\begin{array}{c}0\\0\\ \vdots\\ 1\end{array}\right]\) is the standard basis of \( \mathbb R^m \). Assume that \(\displaystyle L(e_1)=\left[\begin{array}{c} a_{11}\\ a_{21}\\ \vdots \\ a_{n1}\end{array}\right]\), \( \dots \), \(\displaystyle L(e_m)=\left[\begin{array}{c} a_{1m}\\ a_{2m}\\ \vdots \\ a_{nm}\end{array}\right]\). Then \(\displaystyle A=\left[\begin{array}{cccc} a_{11}&a_{12}&\cdots&a_{1m}\\ a_{21}&a_{22}&\cdots&a_{2m}\\ & & \vdots& \\ a_{n1}&a_{n2}&\cdots&a_{nm}\end{array}\right]\) is called the matrix of the transfromation \( L \).

Example 4
 
Determine the matrix of the linear transformation \( L:\mathbb R^3\to\mathbb R^2 \) that satisfies \(\displaystyle L\left(\left[\begin{array}{c}1\\0\\0\end{array}\right]\right)=\left[\begin{array}{c} 3\\2\end{array}\right]\), \(\displaystyle L\left(\left[\begin{array}{c}0\\1\\1\end{array}\right]\right)=\left[\begin{array}{c} 3\\3\end{array}\right]\), and \(\displaystyle L\left(\left[\begin{array}{c}-1\\1\\2\end{array}\right]\right)=\left[\begin{array}{c}1\\4\end{array}\right]\).

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.

Theorem 2
 

Assume that \( L:\mathbb R^m\to\mathbb R^n \) and \( K:\mathbb R^n\to\mathbb R^p \) are two linear transformations. Then the function \( M:\mathbb R^m\to\mathbb R^p \) defined as \( M(u)=K(L(u)) \) for each \( u\in\mathbb R^m \) is a linear transformation.

The following theorem states that the linear combination of two linear transformations with the same domain and codomain is linear.

Theorem 3
 

Assume that \( L:\mathbb R^m\to\mathbb R^n \) and \( K:\mathbb R^m\to\mathbb R^n \) are two linear transformations. If \( \alpha \) and \( \beta \) are two real numbers that the function \( M:\mathbb R^m\to\mathbb R^n \) defined as \( M(u)=\alpha L(u)+\beta K(u) \) for \( u\in \mathbb R^m \) is a linear transformation.

Matrix of the composition. Product of matrices

Theorem 4
 

Assume that \( L:\mathbb R^m\to\mathbb R^n \) and \( K:\mathbb R^n\to\mathbb R^p \) are two linear transformations and that \( Q=K\circ L \). Assume that the \(\displaystyle \hat L= \left[\begin{array}{ccccc} l_{11}&l_{12}&l_{13}& \cdots &l_{1m}\\ l_{21}&l_{22}&l_{23}&\cdots &l_{2m}\\ &&&\vdots&\\ l_{n1}&l_{n2}&l_{n3}&\cdots&l_{nm}\end{array}\right]\), \(\displaystyle \hat K= \left[\begin{array}{ccccc} k_{11}&k_{12}&k_{13}& \cdots &k_{1n}\\ k_{21}&k_{22}&k_{23}&\cdots &k_{2n}\\ &&&\vdots&\\ k_{p1}&k_{p2}&k_{p3}&\cdots&k_{pn}\end{array}\right]\), and \(\displaystyle \hat Q=\left[\begin{array}{ccccc} q_{11}&q_{12}&q_{13}& \cdots &q_{1m}\\ q_{21}&q_{22}&q_{23}&\cdots &q_{2m}\\ &&&\vdots&\\ q_{p1}&q_{p2}&q_{pm}&\cdots&q_{pm}\end{array}\right]\), respectively, are the matrices of \( L \), \( K \), and \( Q \). Then for each \( (i,j)\in\{1,2,\dots p\} \times\{1,2,\dots, m\} \) the following equality holds: \[ q_{ij}=\sum_{s=1}^n k_{is}l_{sj}.\]

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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