by Seth Grief-Albert
A matrix tells you what you can build in a space.
Gaussian Operations
$$ \color{red}{\rho} $$
<aside> 💡 In each row of a system, the first variable with a non-zero coefficient is the row’s leading variable. A system is in echelon form if each leading variable is to the right of the leading variable in the row above it, except for the leading variable in the first row, and any rows with all-zero coefficients are at the bottom.
Row Reduced Echelon Form is the case where the leading ones form a downwards staircase. 📉
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A free loading column does not contain a leading one, and therefore it can take on any value. For each of these, pick a new free variable such as t or s. The goal is to find a vector form of the solution by going through the normal steps and replacing those column vectors with selected parameters.
<aside> 💡 Every leading 1 leads to the image, and every non-leading 1 leads to the kernel.
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$$ n=\dim(Im(L))+\dim(\ker(L))\\ n=Rank(A)+Nullity(A)=\ce{input dim} $$
$$ \ce{If ker(L) is 0-dimensional, all columns contain leading ones} $$
In other words, free variables contribute to the kernel. This can be made sense of by delving into the very idea of free variables; they allow things to not be linearly independent. In other words since everything can be crafted from them, they are arbitrary.
Any linear transform from $\R^n\to\R^m$ can be represented by a $m\times n$ matrix.