Linear Equations

Section 1.1 Linear Equations#

Linear Equations can always be written in the form of (we mainly use this form in linear algebra):

Where $b$ and the coefficients $a_1,\ldots,a_n$ are $\in \mathbb{R}$ or $\mathbb{C}$.

A system of linear equations is simply a collection of linear equations involving the same variables (shared across all equations). A solution to the system is an ordered list $(s_1,s_2,\ldots,s_n)$ of numbers that make each equation true when $s_i$ is substituted in for the respective variable $x_i$.

Sidenote: Set of all possible solutions is called a solution set. It can be empty (no solution), a single point (unique solution, dimension 0), or infinite: a line (dimension 1), a plane (dimension 2), or higher dimensional analogs.

We will see later that the dimension of the solution set (assuming there is a solution) equals the number of "free variables" a system has. But whether or not there exists a unique solution, or infinitely many, has no effect on the structure of the mentioned solution list that we defined above. Each solution is still an ordered list of exactly $n$ scalars, one per variable.

All a higher-dimensional solution set means, is that there are more of these lists (slight oversimplification), forming a larger shape inside whatever space we're working in. This distinction between solution set and solution list is important. So we can think of free variables as "knobs" that may or may not appear according to the system we work with that allow us to select different solutions out of that set. But we shall understand and see more clearly about all of these ideas when we get there!

Python code blocks are executable#

import sys
print(f"Python {sys.version}")
print("Hello from Pyodide!")