A function \(L\colon \R^2\to \R^2\) is called a linear mapping (or linear map) if \(L\) respects the operations of vector addition and scalar multiplication, that is, if
for all \({\mathbf u},{\mathbf v}\in \R^2\) and all \(\alpha\in \R\text{.}\) It is customary to omit the parentheses in the expression \(L({\mathbf u})\) and simply write \(L{\mathbf u}\) to denote the value of a linear mapping. With this convention, equations (2.2.1) and (2.2.2) become
For each of the following functions, determine whether or not it is linear. If a given function is linear, show that it satisfies properties (2.2.1) and (2.2.2). If a given function is not linear, give a specific example to show that it fails at least one of the properties (2.2.1) or (2.2.2).
the function that sends every input point to its reflection across the \(x\)-axis
Thus, a linear mapping has a particularly simple form, and is summarized by four real constants \(a_1,b_1,a_2,b_2\) that are the components of the two values \(L{\mathbf e}_1,L{\mathbf e}_2\text{.}\) The square array of numbers \(\twotwo{a_1}{a_2}{b_1}{b_2}\) is called the matrix for \(L\text{,}\) denoted \([L]\). The array \(\begin{bmatrix}
x\\y\end{bmatrix}\) is called the column vector that represents \({\mathbf u}=(x,y)\text{.}\) Equation (2.2.6) motivates the following definition of the operation that multiplies a matrix times a column vector.
If the matrix for \(L\) is \(\twotwo{a}{b}{c}{d}\text{,}\) and the matrix for \(M\) is \(\twotwo{e}{f}{g}{h}\text{,}\) then the matrices (the plural of matrix is matrices) for the linear maps \(\alpha L\text{,}\)\(L+M\text{,}\) and \(LM\) are given by the following.
\begin{align}
(\text{matrix for } \alpha L) \amp = \twotwo{\alpha a}{\alpha b}{\alpha c}{\alpha d}\tag{2.2.11}\\
(\text{matrix for } L+M) \amp = \twotwo{a+e}{b+f}{c+g}{d+h}\tag{2.2.12}\\
(\text{matrix for } LM) \amp = \twotwo{ae+bg}{af+bh}{ce+dg}{cf+dh}\tag{2.2.13}
\end{align}
These natural constructions motivate the following matrix operations, called matrix scalar multiplication, matrix addition, and matrix multiplication, respectively.
Notice that every entry in the matrix on the right side of (2.2.16) is a dot product. For example, the entry \(af+bh\) in the upper right corner (top row, right column) of the matrix on the right side of (2.2.16) is the dot product
of the top row of \(\twotwo{a}{b}{c}{d}\) with the right column of \(\twotwo{e}{f}{g}{h}\text{.}\) The row and column matching pattern works for every entry in the product matrix.
Choose random integer values between \(-5\) and \(+5\) for the constants \(\alpha,a,b,c,d,e,f,g,h,x,y\text{.}\) Set \(S=\twotwo{a}{b}{c}{d}\text{,}\) let \(T=\twotwo{e}{f}{g}{h}\text{,}\) and let \(\mathbf{u}=\begin{bmatrix}x\\y\end{bmatrix}\text{.}\) Calculate \(\alpha
S\text{,}\)\(S+T\text{,}\)\(ST\text{,}\)\(TS\text{,}\)\(S\mathbf{u}\text{,}\) and \(T\mathbf{u}\text{.}\) Check your answers with an online matrix algebra calculator. Repeat the exercise until you can do matrix multiplication quickly and accurately, without having to look at the formula (2.2.16).
Find a set of values for \(a,b,c,d,e,f,g,h\) for which \(ST\neq TS\text{.}\) Find another set of values for \(a,b,c,d,e,f,g,h\text{,}\) all of them nonzero, for which \(ST=TS\text{.}\)
The identity function \(\Id\colon \R^2\to\R^2\) is linear, and the matrix \(\twotwo{1}{0}{0}{1}\) for the identity function is called the identity matrix. It is easy to check that
If \(L,M\) are linear mappings such that \([L] =
\twotwo{a}{b}{c}{d}\) and \([M] =
\frac{1}{ad-bc}\twotwo{d}{-b}{-c}{a}\text{,}\) then (2.2.18) becomes
\begin{equation}
L\of M = M\of L = \Id \tag{2.2.21}
\end{equation}
and we conclude that \(L,M\) are inverses to one another. This shows that, if \([L]=\twotwo{a}{b}{c}{d}\) and \(ad-bc\neq 0\text{,}\) then \([L^{-1}]=\frac{1}{ad-bc}\twotwo{d}{-b}{-c}{a}\text{,}\) and motivates the following definition of the inverse of a matrix.
Comment: The quantity \(ad-bc\) is called the determinant of the matrix \(\twotwo{a}{b}{c}{d}\text{,}\) denoted \(\det\left(\twotwo{a}{b}{c}{d}\right)\).
Verify that the identity function \(\Id\colon \R^2\to \R^2\) is linear, and that the matrix for the identity function is \(\twotwo{1}{0}{0}{1}\text{.}\)
Explain each of the connecting phrases “which is the same as”, “it follows that”, and “we conclude that” immediately following equations (2.2.19), (2.2.20), and (2.2.21). Hint.
To obtain the matrix for a linear map \(L\text{,}\) use the fact that the first column of the matrix is the column vector \(L\mathbf{e}_1\text{,}\) and the second column is the column vector \(L\mathbf{e}_2\text{.}\)
\begin{align}
ax + by \amp = r\tag{2.2.24}\\
cx + dy \amp = s\notag
\end{align}
where \(a,b,c,d,r,s\) are constants, and \(x,y\) are unknowns, is called a linear system of equations in two unknowns. A solution is a pair of values for \(x,y\) so that both equations in (2.2.24) hold. We can use matrix algebra to solve linear systems of equations, as follows. Using the notation defined in (2.2.7), here is the matrix version of (2.2.24).
The matrix \(A=\twotwo{a}{b}{c}{d}\) is called the matrix of coefficients for the linear system. If we write \(\mathbf{u}=\begin{bmatrix}x\\y\end{bmatrix}\) and \(\mathbf{v}=\begin{bmatrix}r\\s\end{bmatrix}\) then we can write (2.2.25) even more compactly as
Choose random integer values between \(-5\) and \(+5\) for the constants \(a,b,c,d,r,s\text{.}\) Let \(A=\twotwo{a}{b}{c}{d}\text{,}\) let \(\mathbf{u}=\begin{bmatrix}x\\y\end{bmatrix}\text{,}\) and let \(\mathbf{v}=\begin{bmatrix}r\\s\end{bmatrix}\text{.}\) Solve the linear system \(A\mathbf{u}=\mathbf{v}\) for \(\mathbf{u}\text{.}\) Check your answers with an online matrix algebra calculator. Repeat the exercise until you can do matrix inversion and solve the system quickly and accurately, without having to look at the formula (2.2.22).
Let \(\ell\) be a line through the origin. Show that reflection \(F_\ell\) across the line \(\ell\) can be written as a composition \(R_{\theta}\of F_X \of R_{-\theta}\) for some appropriate value of \(\theta\text{.}\) Find the matrix for \(F_\ell\text{.}\)
