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GEM

\(A\ \underrightarrow{\text{ Elementary Row Operations }}\ \text{Reduced Row Echelon Form (RREF)}\)

Matrix is in RREF if - All zero rows are at the bottom - The first non-zero element of each row is 1, and all other entries in its column are 0 - The first non-zero element of each row occurs in a column to the right of the first non-zero entry in the preceding rows

Theorem 3.15

Let \(Ax=b\) be system with \(r\) non-zero equations in n unknowns. Suppose \(\text{Rank}(A)=\text{Rank}(A|B)\) and \((A|B)\) is in RREF. Then:

  • \(\text{Rank}(A)=r\)
  • if a general solution is obtained via GEM of the form, \(s=s_0+t_1v_1+...+t_{n-r}v_{n-r}\) then, \(\{v_i\}\) is a basis

Prop

Elementary Row Operations on an augmented matrix give an equivalent system.