Linear Algebra (Part-1): 2-D Real Coordinate Space
PREFACE
This is the beginning of a new series on Linear Algebra. Throughout this series, we will cover Linear Algebra at an undergraduate level, useful both generally and for Machine Learning. This series is not for absolute beginners; hence, you should be at least familiar with Linear Algebra at the high school level. I will share my learnings and understanding of Linear Algebra, and I hope you gain a good understanding of the concepts and find value in this series. Be sure to engage in the comments for discussions to make this more interactive, and I would love to be a part of those discussions. Let’s begin the journey!
2-D Real Coordinate Space
What is a 2-D Real Coordinate Space?
The formal definition of 2-D Real Coordinate Space is:
A set of all possible real-valued two tuples is called a 2-D Real Coordinate Space.
But what is a 2-tuple?
A tuple is an ordered list of numbers and a 2-tuple is an ordered list of two numbers.
So, a 2-D Real Coordinate Space can be thought of as selecting all the possible combinations of two numbers from the set of Real Numbers. This is indeed true, for example, the 2-D Coordinate Space that we are used to has points that are combinations of Real Numbers. Look at the figure below to visualize the 2-D Real Coordinate Space.
Remember that “Space” here refers to a mathematical space and not to a physical space.
A real coordinate space is represented by ℝⁿ and n represents the dimension of the space. Hence ℝⁿ represents an n-dimensional real coordinate space. Here we are dealing with a 2-D coordinate space so n is 2, hence a 2-D coordinate space is represented as ℝ².
Parametric Representation Of Lines
Consider a line L in a 2-D real coordinate space as shown in the figure below.
Note: Vectors are represented in bold italics.
In the figure above, L is a line and tv is a group of an infinite number of vectors colinear to each other and parallel to the line L. Adding a vector x to tv lands us on a point on the line L using the triangle law of vector addition.
So, the parametric equation of the line L can be written as,
L = { x + tv | t ∈ ℝ }
Understanding this equation is fairly simple, for every possible value of ‘t’ in the set of real numbers we have a vector tv and adding a constant vector x to tv lands us on a point on the line L. So for every possible value of ‘t’ we land on a point on Line L and in this way, we can land on every point on the line.
This definition works in any dimension.
Understanding the parametric equation of a line will help us to understand the equation of a plane in n dimensions. That’s all for the first part, I hope that you enjoyed learning with me.
