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@jemoka / Jemoka Knowledge Base / wiki/concepts/linear_systems.md
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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’s either a unique solution, no solution, infinite solutions— problems with zero “zero” 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—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 & 2 \\ 1e-4 & 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 & 2 \\ 1 & 0) \mqty(c_3 \\ c_2) = \mqty(\ddots) \end{equation} the right side needn’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) — see singular matrix: one column is linearly dependent on the others determinant is 0 non-empty null space Diagonal Matrix see Diagonal Matrix