A tuple is a column of mathematical objects. (Some books define a tuple as a row of mathematical objects.) If the tuple has \(n\) entries (or components), we call it an \(n\)-tuple. \(n\) is also called the size of the tuple. The following are all \(3\)-tuples: \[ \begin{bmatrix} \frac{1}{5} \\ \sqrt{3}-2 \\ 2^9 \end{bmatrix},~ \begin{bmatrix}-2 \\ 3 \\0\end{bmatrix},~ \begin{bmatrix} x^2-1 \\ \frac{1}{\sqrt{2}}x \\ x+\pi \end{bmatrix}. \] The last one is a tuple of polynomials in \(x\).
Here is a \(2\)-tuple: \(\begin{bmatrix} 2\\-1\end{bmatrix}\).
Here ia a \(5\)-tuple: \(\begin{bmatrix} 1 \\ 5 \\ 0 \\-3 \\ -7\end{bmatrix}\).
Given an \(n\)-tuple \(t\), the \(i\)th entry of \(t\) is given by \(t_i\). For example, if \(t = \begin{bmatrix} 1/5 \\ \sqrt{3}-2 \\ 2^9 \end{bmatrix}\), then \(t_1 = 1/5\), \(t_2 = \sqrt{3}-2\), and \(t_3 = 2^9\).
If addition and multiplication can be performed on the entries in a tuple, then addition and scalar multiplication on tuples can be defined.
To add two tuples having the same size, simply add the corresponding entries when forming the result. For example, \[\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}+ \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} = \begin{bmatrix} 1+4 \\ 2+5 \\ 3+6 \end{bmatrix} = \begin{bmatrix} 5 \\ 7 \\ 9 \end{bmatrix}.\]
Let \(t\) be an \(n\)-tuple and \(\alpha\) be a number. Then \(\alpha t\) is an \(n\)-tuple obtained from \(t\) by multiplying each entry by \(\alpha\). For example, \[2~\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}= \begin{bmatrix} 2\times 1 \\ 2\times 2 \\ 2\times 3 \end{bmatrix}= \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix},\] and \[\sqrt{2}~\begin{bmatrix} \sqrt{2} \\ 1/\sqrt{2} \end{bmatrix}= \begin{bmatrix} 2 \\ 1 \end{bmatrix}.\]
Note that if \(u\) and \(v\) are \(n\)-tuples of numbers, \(u-v\) is defined to be \(u + (-1)v\). This gives the notion of tuple subtraction. For example, \[\begin{bmatrix} 1 \\ 2 \end{bmatrix} -\begin{bmatrix} 2 \\ -1 \end{bmatrix}= \begin{bmatrix} 1 \\ 2 \end{bmatrix} +\begin{bmatrix} -2 \\ 1 \end{bmatrix}= \begin{bmatrix} -1 \\ 3 \end{bmatrix}.\] Quick Quiz
Compute \(\begin{bmatrix} 2 \\ 3 \\ 1 \\ 0 \end{bmatrix}- \begin{bmatrix} 0 \\ 1 \\ 4 \\ 0 \end{bmatrix}+ 2~\begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix}.\)