Exact Linear Algebra for SAGE - William Stein - University of ...
Exact Linear Algebra for SAGE - William Stein - University of ...
Exact Linear Algebra for SAGE - William Stein - University of ...
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Echelon Forms <strong>of</strong> Matrices<br />
Computing Echelon Forms<br />
Decomposing Spaces Under the Action <strong>of</strong> Matrix<br />
Reduced Row Echelon Form<br />
Definition (Reduced Row Echelon Form)<br />
A matrix is in (reduced row) echelon <strong>for</strong>m if each row in the matrix has<br />
more zeros at the beginning than the row above it, the first nonzero entry <strong>of</strong><br />
every row is 1, and the first nonzero entry <strong>of</strong> any row is the only nonzero<br />
entry in its column.<br />
Given a matrix A, there is another matrix E such that E is obtained<br />
from A by left multiplication by an invertible matrix and E is in reduced<br />
row echelon <strong>for</strong>m.<br />
This matrix E is called the echelon <strong>for</strong>m <strong>of</strong> A. It is unique.<br />
A pivot column <strong>of</strong> A is a column <strong>of</strong> A such that the reduced row echelon<br />
<strong>for</strong>m <strong>of</strong> A contains a leading 1.<br />
<strong>William</strong> <strong>Stein</strong> <strong>Exact</strong> <strong>Linear</strong> <strong>Algebra</strong> <strong>for</strong> <strong>SAGE</strong>
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