We prove the Rank Nullity Theorem. This theorem states that for any linear operator (matrix) which maps one finite dimensional vector space to another that the rank of the matrix plus the dimension of the null space is the dimension of the domain (number of columns). In matrix lingo, it asserts that the number of free variables plus the number of leading ones is the number of columns. The technique of this proof will be used over and over again in linear algebra, so it is very important to understand the basis extension method.
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The Rank Nullity Theorem
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mike the mathematicianmike dabkowski mathlinear alegbralinear equationssystems of equationssolutions to systems of linear equationssystems of linear equationsmath 227 umichmath 217 umichmatrix algebraechelon formreduced row echelon formrank of matrixnull space of matrixrank nullity theoremrank nullity theorem proofrank nullity for non square matrixrank nullityrank nullity formularank nullity linear algebrarank nullity for matrix
