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is really just finding the right "mix" of columns to reach a target point in space. The Heart of the Matter: lecture notes for linear algebra gilbert strang
Properties of det, eigenvalues, eigenvectors, diagonalization. : is really just finding the right "mix"
In Strang’s hands, the equation $\textdim(Row Space) + \textdim(Nullspace) = n$ (the Rank-Nullity Theorem) becomes a law of conservation. It teaches the student that every linear transformation preserves a certain amount of information (the rank) and discards the rest (the nullity). The matrix is no longer just a grid; it is a filter, straining out specific dimensions of reality while preserving others. It teaches the student that every linear transformation
: Solve ([A \ | \ I] \rightarrow [I \ | \ A^-1]) by elimination.
His famous opening line in the 18.06 lectures is: “The fundamental problem of linear algebra is to solve a system of linear equations.” But he doesn't stop there. He immediately introduces the —the idea that solving ( Ax = b ) is about finding the right combination of the columns of ( A ).