Suggest edit — Linear Systems

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---
title: "Linear Systems"
type: concept
related: [Invertability, Null Space, Diagonal Matrix, Linear Map]
source: https://www.jemoka.com/posts/kbhlinear_systems/
confidence: high
status: active
---

Systems of Linear Equations \begin{equation} T v = v' \end{equation}
every system of linear equations is decomposed into this. Classically, there&rsquo;s either a unique solution, no solution, infinite solutions&mdash;
problems with zero &ldquo;zero&rdquo; is really hard to define. For instance:
\begin{equation} 6.23423 \times 10^{192} - 1 \times 10^{7} = 6.23423 \times 10^{192} \end{equation}
so in this case \(10^{7}\) literally behaves like zero. (small numbers have the opposite problem)
so, we use elementary row operations to make sure that enormous numbers are essentially standardized&mdash;if a row has huge numbers, we may want to scale it down to smaller numbers to make them nice.
row scaling scaling an entire row by multiplying the number a la elementary row operations
column scaling scaling a column by changing the definition of \(c_{j}\); for instance,
\begin{equation} \mqty(3e-4 &amp; 2 \\ 1e-4 &amp; 0) \mqty(c_1 \\ c_2) = \mqty(\ddots) \end{equation}
we can set \(c_3 = c_1(1e-4)\) and write
\begin{equation} \mqty(3 &amp; 2 \\ 1 &amp; 0) \mqty(c_3 \\ c_2) = \mqty(\ddots) \end{equation}
the right side needn&rsquo;t to get scaled since we simply changed the definition of \(x\).
square matrix a square matrix is a invertable Linear Map.
solvability singular matrix (non-solvable matrix) &mdash; see singular matrix:
one column is linearly dependent on the others determinant is 0 non-empty null space Diagonal Matrix see Diagonal Matrix