A record becomes a vector

Suppose we record study hours, completed practice sets, and a quiz score. These are invented numbers for working through the arithmetic:

Record	Hours	Sets	Score
1	1	0	2
2	2	1	5
3	3	1	8

The first two measurements are inputs. The score is the output we want to predict. Keeping that distinction matters: putting the score among the inputs would give the model information it will not have when making a new prediction.

An ordered list of measurements

Record 2 has input measurements 2 and 1. Put them in a column:

$$x_2=\begin{bmatrix}2\\1\end{bmatrix}.$$

This is a vector. Here it represents one record, with hours first and sets second. Swapping the entries changes its meaning. Calling it a vector does not make the measurements more abstract; it gives us a way to calculate with them together.

The subscript in $x_2$ identifies the second record. To name a single entry, use $x_{ij}$: record $i$, feature $j$. Thus $x_{21}=2$ hours and $x_{22}=1$ set. Different books choose different conventions, so check the definition before interpreting a subscript.

The notation $x_i\in\mathbb R^2$ says that each input has two real-number entries. Read $\in$ as “belongs to” and $\mathbb R^2$ as “the collection of real vectors with two entries.” The superscript describes the number of coordinates; it does not square the measurements.

Calculating entry by entry

Subtract records 1 and 2 in matching positions:

$$x_2-x_1= \begin{bmatrix}2-1\\1-0\end{bmatrix} =\begin{bmatrix}1\\1\end{bmatrix}.$$

Record 2 contains one more hour and one more set. Vector subtraction keeps both differences. Multiplying a vector by a single number, called a scalar, scales every entry: $2x_2=[4,2]^\mathsf T$. The superscript $\mathsf T$ turns a displayed row into a column; the next lessons explain this operation, the transpose.

You can also plot $x_2$ at horizontal coordinate 2 and vertical coordinate 1. The plot and column are two representations of the same ordered pair. With hundreds of features, the arithmetic remains available even though a two-dimensional plot cannot show every coordinate.

Units still matter. Converting hours to minutes changes the first coordinate from 2 to 120. It does not change the underlying record. Raw numerical distance between records therefore depends on how their features are measured.

Try it

What is $x_3-x_2$, and what does each entry mean?

Work through the answer

$[3-2,1-1]^\mathsf T=[1,0]^\mathsf T$: record 3 has one more study hour and the same number of practice sets. This comparison does not establish that the extra hour caused the score difference.