Vectors serve as the primary bridge between physical observations and algebraic computation in data science.
To a computer, a vector is an ordered one-dimensional array of numerical values.
To a mathematician, it is an element of a vector space that can be added and scaled according to axioms.
To a visual thinker, a two-dimensional vector points to a coordinate pair on a flat Cartesian plane.
In a dataset, each individual row can be viewed as an observation vector spanning multiple feature dimensions.
Similarly, an entire column can be viewed as a feature vector containing measurements across all observed rows.
Vectors can be arranged vertically as column vectors or horizontally as row vectors.
By default, linear algebra literature treats individual vectors as column vectors unless specified otherwise.
The length or dimension of a vector indicates how many distinct numerical measurements it carries.
When two observations have matching dimensions, algorithms can measure the distance and angle between them.
A crucial rule is that vector operations require matching dimensionality across the participating vectors.
You cannot add a three-dimensional vector directly to a five-dimensional vector in standard linear algebra.
Beginners often confuse the mathematical dimension of a vector with the number of rows in a whole dataset.
Feature dimensions refer to the attributes measured on each single entity rather than the total count of entities.
Treating observations as vectors unlocks the geometric and algebraic tools used throughout data science.