# 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.

## Practice problems

Problem 1. Assume that $$L:\mathbb R^2\to\mathbb R^2$$ is a linear operator such that $$\displaystyle L\left[\begin{array}{c} 1\\ -1\end{array}\right]=\left[\begin{array}{c}6\\ -1\end{array}\right]$$ and $$\displaystyle L\left[\begin{array}{c} 2\\ -3\end{array}\right]=\left[\begin{array}{c}7\\ 5\end{array}\right]$$. If $$\displaystyle L\left[\begin{array}{c} 17\\ -24\end{array}\right]=\left[\begin{array}{c} x\\ y\end{array}\right]$$, determine $$x+y$$.

Problem 2. Assume that $$L:\mathbb R^{9}\to\mathbb R^{7}$$ and $$K:\mathbb R^{7}\to\mathbb R^{9}$$ are two linear transformations. Let $$M=L\circ K$$. How many rows does the matrix of the transformation $$M$$ have?

Problem 3. Given two matrices, $$\displaystyle A=\left[\begin{array}{ccc} 3& 2& -7\\ 5& 7& -11\\ -3& -8& 9\end{array}\right]$$ and $$\displaystyle B=\left[\begin{array}{ccc} 4& -7& 6\\ -5& -15& 9\\ -5& 7& -5\end{array}\right]$$, let $$C=A\cdot B$$. If $$\displaystyle C=\left[\begin{array}{ccc} {c_{11}}& {c_{12}}& {c_{13}}\\ {c_{21}}& {c_{22}}& {c_{23}}\\ {c_{31}}& {c_{32}}& {c_{33}}\end{array}\right]$$, determine $$c_{23}$$.

Problem 4. Assume that $$L:\mathbb R^{3}\to\mathbb R^{2}$$ and $$K:\mathbb R^{2}\to\mathbb R^{4}$$ are two linear transformations. Let $$M=K\circ L$$. If the matrix of $$L$$ is $$\displaystyle \left[\begin{array}{ccc} 3 &-2 &6 \\ -5 &-3 &2 \end{array}\right]$$, the matrix of $$K$$ is $$\displaystyle \left[\begin{array}{cc} 9 &-2\\ 4 &-3\\ -5 &3\\ -9 &7 \end{array}\right]$$, and the matrix of $$M$$ is $$\displaystyle \left[\begin{array}{ccc} m_{11} &m_{12}&m_{13}\\ m_{21} &m_{22}&m_{23}\\ m_{31} &m_{32}&m_{33}\\ m_{41} &m_{42}&m_{43}\\ \end{array}\right]$$, what is the value of $$m_{11}+m_{22}+m_{33}-m_{41}-m_{42}-m_{43}$$?

Problem 5. Assume that $$L:\mathbb R^3\to\mathbb R^2$$ is a linear transformation such that $$\displaystyle L\left[\begin{array}{c} -1\\ 2\\1\end{array}\right]=\left[\begin{array}{c}-11\\ 30\end{array}\right]$$, $$\displaystyle L\left[\begin{array}{c} 1\\ 0\\1\end{array}\right]=\left[\begin{array}{c}-1\\ -10\end{array}\right]$$, and $$\displaystyle L\left[\begin{array}{c} 0\\ 1\\2\end{array}\right]=\left[\begin{array}{c}-14\\ 3\end{array}\right]$$. If $$\displaystyle \left[\begin{array}{ccc} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\end{array}\right]$$ is the matrix of the transformation $$L$$, determine $$a_{11}+a_{22}+a_{13}$$.