Year 1 · Week 03
Chapter 3: Linear Equations and Coordinate Systems
In Week 1, we discovered something surprising: the y-axis doesn't have to be perpendicular to the x-axis! We drew "wacky" coordinate systems where the axes meet at different angles. This week, we'll use systems of linear equations to understand exactly how these coordinate systems work.
Part 1: A Quick Puzzle
Let's start with a simple system of equations:
x + y = 5
y = 5
What are x and y?
Since the second equation tells us y = 5, we can substitute that into the first equation:
x + 5 = 5
x = 0
So the solution is x = 0, y = 5, or the point (0, 5).
Part 2: Another Puzzle
Now let's try a slightly different system:
x + y = 5
y = 3
Again, we substitute y = 3 into the first equation:
x + 3 = 5
x = 2
The solution is x = 2, y = 3, or the point (2, 3).
Part 3: Linear Algebra Notation
Mathematicians have developed a compact way to write coordinate transformations using matrices. Instead of solving systems of equations each time, we can use matrix multiplication!
A matrix is a rectangular array of numbers. When we multiply a matrix by a vector, we can transform points between coordinate systems.
This is how robots and computer graphics handle coordinate transformations—everything uses matrix math!
Key Takeaways This Week
- Systems of equations help us find where two lines intersect
- Coordinates don't have to be perpendicular—we can use "wacky" coordinate systems
- Linear algebra (matrices) makes coordinate transformations easier
- These concepts are essential for robot navigation and computer graphics