Ordered Tuples as Vectors in Rn

What an Ordered Tuple Represents

A vector in \(\mathbb{R}^n\) is described by \(n\) real numbers placed in a specified order. The order matters: the first entry has a first position, the second entry has a second position, and so on through position \(n\). The notation records both the numerical entries and their positions. This makes it possible to refer precisely to, for example, the fourth coordinate of a vector without confusing its location with its value.

Throughout this tutorial, \(n\) is a positive integer and the scalars are real numbers. In keeping with the course convention, vectors are columns. We use the notation \((x_1,\dots,x_n)^{\top}\) for a column whose entries, from top to bottom, are \(x_1,\dots,x_n\). The subscript on \(x_i\) labels the position: \(x_1\) is in position 1, \(x_2\) in position 2, and \(x_n\) in position \(n\).

An ordered tuple can also be described as a rule assigning a real number to each position in the index set \(\{1,\dots,n\}\). The value assigned to position \(i\) is \(x_i\). This description explains why subscripts are useful: they specify positions in the list. It also explains why \(n\) is not an entry of the vector just because it appears in the notation \(\mathbb{R}^n\). The superscript \(n\) tells us how many coordinate positions there are; the entries are the numbers \(x_1,\dots,x_n\).

The word ordered is essential. A coordinate list is not an unordered collection: the place where a number occurs is part of how we describe the vector. Nor does the value of a coordinate determine its position. A coordinate may be negative, positive, zero, or repeated elsewhere in the list. The position is specified by its index, not by the size or sign of its value.

Counting and Locating Coordinates

The coordinate positions of \(\mathbf{x}\in\mathbb{R}^n\) are indexed by the integers \(1,2,\dots,n\). The first coordinate is \(x_1\), and the last is \(x_n\). The notation \(x_i\) is used only when \(i\) is an allowed position, so \(1\leq i\leq n\). When the vector is displayed as a column, the position and the row match: the entry in row \(i\) is coordinate \(x_i\).

Proof. By definition, an ordered \(n\)-tuple has one entry assigned to each index in \(\{1,\dots,n\}\), and has no coordinate positions indexed outside that set. This index set contains exactly the \(n\) integers \(1,\dots,n\). Therefore the tuple has exactly \(n\) coordinate positions. \(\square\)

This count does not depend on the values in the tuple. For instance, a list with three entries still has three coordinate positions if two entries are the same, and it still has three if one or more entries are zero. The coordinate dimension counts positions, not distinct numerical values.

Proof. By the definition of \(\mathbf{x}\), its entries are assigned in order to the positions \(1,\dots,n\), with \(x_i\) assigned to position \(i\). Since \(k\) is in this range, position \(k\) exists and its assigned entry is \(x_k\). Thus reading position \(k\) returns \(x_k\). \(\square\)

The proposition is simple, but its discipline is useful: first identify the requested position, then read the entry carrying that subscript. For example, if the question asks for coordinate 3, find the third entry. Do not use the numerical value of the index as the answer, and do not count from zero.

Worked Examples: Reading Entries and Identifying Dimension

Using the Notation Carefully

Three pieces of notation work together. The symbol \(\mathbb{R}^n\) specifies the coordinate space, the list gives the entries in order, and a subscript identifies one entry. For example, in \(\mathbf{q}=(2,-5,8)^{\top}\), the exponent \(3\) in the space \(\mathbb{R}^3\) would indicate three coordinate positions, while \(q_2\) would refer to the second entry, \(-5\). Neither the superscript nor the subscript is itself an entry of the vector.

To identify the coordinate dimension from a display, count entries rather than inspecting their values. A column with two entries has two coordinate positions; a column with five entries has five. To read a coordinate, use the index as a position label. The first entry has subscript 1, even if its value is 100, and the fourth entry has subscript 4, even if its value is \(-\frac{1}{3}\).

In this tutorial, “coordinate dimension” refers to the number of real coordinate positions specified by \(n\) in \(\mathbb{R}^n\). This concrete usage answers the questions “How many entries are there?” and “Which coordinate is being named?” The entries themselves can be any real numbers; no extra restriction is imposed on them by the notation alone.

When a vector is written in compact tuple notation, the superscript \(\top\) indicates that the listed entries are arranged as a column. For example, \((1,0,-2)^{\top}\) means the column with \(1\) at the top, \(0\) beneath it, and \(-2\) at the bottom. This convention keeps coordinate lists compact while respecting the rule that vectors in \(\mathbb{R}^n\) are columns.

Common Mistake: Confusing an Index with a Value

Suppose \(\mathbf{u}=(8,-3,4)^{\top}\). A mistaken reading is “\(u_2=2\), because the subscript is 2.” This treats the position label as though it were the coordinate value. The correction is to use the subscript to locate the entry: position 2 contains \(-3\), so \(u_2=-3\). The value \(2\) is not an entry of this vector.

A related counting error is to call a three-entry column an element of \(\mathbb{R}^2\) because it contains two commas. Commas separate entries; they are not coordinate positions. The column \((8,-3,4)^{\top}\) has three entries and therefore belongs to \(\mathbb{R}^3\), not \(\mathbb{R}^2\). Counting the entries directly avoids that error.

It is also possible to read a column from the wrong end. In \((8,-3,4)^{\top}\), the first coordinate is the top entry, \(8\), not the bottom entry. Coordinate indices start at \(1\) at the top and increase by one as we move down the column. Keeping that direction fixed makes coordinate references consistent.