Linear algebra

Linear algebra formalizes and extends matrix arithmetic: a vector's magnitude (its length) comes from the Pythagorean theorem, and the dot product of two vectors, multiply matching entries and add, measures how much they point the same direction: (2,3) dot (4,1) = 2(4) + 3(1) = 11 (a dot product of 0 means the vectors are perpendicular). Beyond just knowing matrices can represent a system of equations, a 2x2 matrix [[a,b],[c,d]] has a determinant, ad minus bc, that reveals whether a system has a unique solution ([[2,1],[1,3]] has determinant 2(3) - 1(1) = 5, a nonzero value guaranteeing one), and Gaussian elimination is the actual step by step method for finding it, turning the system into an equivalent triangular one by adding multiples of one row to another.

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