Step 3a: Swap the 3rd row with a lower one so that the leftmost nonzero entry is in the 3rd row.Step 2c: Use row replacement so all entries below this 1 are 0.Step 2b: Scale the 2nd row so that its first nonzero entry is equal to 1.Step 2a: Swap the 2nd row with a lower one so that the leftmost nonzero entry is in the 2nd row.Step 1c: Use row replacement so all entries below this 1 are 0.Step 1b: Scale the 1st row so that its first nonzero entry is equal to 1.Step 1a: Swap the 1st row with a lower one so a leftmost nonzero entry is in the 1st row (if necessary).We will not prove uniqueness, but maybe you can! Algorithm (Row Reduction) This assumes, of course, that you only do the three legal row operations, and you don’t make any arithmetic errors. The uniqueness statement is interesting-it means that, no matter how you row reduce, you always get the same matrix in reduced row echelon form. We will give an algorithm, called row reduction or Gaussian elimination, which demonstrates that every matrix is row equivalent to at least one matrix in reduced row echelon form. Subsection 1.2.3 The Row Reduction Algorithm TheoremĮvery matrix is row equivalent to one and only one matrix in reduced row echelon form. This is one possible explanation for the terminology “pivot”. We used the pivot position in the matrix in order to make the blue line pivot like this. We can think of the blue line as rotating, or pivoting, around the solution ( 1,1 ). What has happened geometrically is that the original blue line has been replaced with the new blue line y = 1.
0 Comments
Leave a Reply. |