The Definition of Linear Space

Space Imagined Linearly

• Mathematics offers us a way to define linear space, but picturing linear space is very easy to do. To begin understanding linear space, imagine a blank piece of paper. Now imagine a line drawn on the paper. This line is a representation of linear space. Basically, it is a line and it is taking up space. In math, this line is likely to be referred to as a "vector." The only difference between a vector and a line is that a vector has defined direction and size.

Creating Linear Space Mathematically

• Linear space is represented in mathematics via various equations. A very simple example of a linear equation is "x=y." By plugging any number into "x," an equivalent "y" value is produced. On a standard linear graph with an x and a y-axis, this equation would be represented by a single diagonal line. At any point on the line, the x value and the y value would be equal. In this example, all of linear space is made up of this single line. By modifying the equation and adding other variables, lines can be made more complex, limited in length or have their shape changed.

Usefulness of Linear Space

• Linear space is useful in the field of mathematics because it provides a stable, predictable model for a variety of variables. By using a linear equation to plot a line, a mathematician can view every possible outcome. For example, if someone is trying to calculate future profits. For every one item sold, there is a $5.00 profit. By using "x" to represent sales, all future profits are predicted based on sales by making the other side of the equation "y x 5" or "5y." By creating this line on a graph, it is possible to view profits for any number of sales by following the line to the point where "x" equals future sales. The "y" value at this point will show you what profits would be at this point. Of course, this is just a very basic example. More complicated representations of linear space are possible with further studies of mathematics.

Linear Space in Real Life

• It is likely that you encounter linear space every day. Many two dimensional images or representations of objects can be, at least partly, existing in linear space. Many digital designs use vector graphics to create characters and logos. Just like in math, the vectors in this art refers to the lines that comprise the image. These vectors are arranged by the artist in specific ways to evoke an image. If needed, these vectors could be explained with a series of very complex linear equations, but this level of understanding is not necessary to the artist as a computer program typically handles the manipulation of vectors in this kind of vector art.