Due to closure, every vector in a vector space is a linear combination of other vectors in the space.
A set of vectors that can be used to form linear combinations that generate other vectors is called a spanning set of vectors. The vectors that can be created by those vectors is called the span of those vectors.
If a set of vectors in a vector space can be used to form linear combinations that generate all vectors of the vector space, that set of vectors is a spanning set for the vector space, with the vector space as the span of that spanning set of vectors.
If the spanning set of a vector space is the fewest possible number of vectors that can span the space (in our examples so far, two vectors), that number of vectors is the number of dimensions of the vector space.
Consider if we tried to use only one vector as a spanning set in the examples so far. A linear combination to create a second vector in the vector space would only have one vector in the linear combination, without any other vector added to it. Hence, the linear combination would only perform scalar multiplication, which would not allow any other lines to be generated. All scalar multiples of a line are on the same line. That is a single dimension: spanning only a line.
In the examples of using coefficients of a line equation as vectors, a single dimension would be to have only one line, for example only having the x-axis without a y-axis, or only having a y-axis with no x-axis, etc.
In fact, the x and y axes are two lines that together form a spanning set that spans the plane. This will be covered in analytic geometry, which uses graph coordinates as vector components, instead of using line equation coefficients as vector components which we have been doing so far.
Before embarking on a journey into analytic geometry, we mention that we will return to using coefficients of a line equation as vector components later, because many phenomena that need to be modeled behave like line equations.