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| lecture_4 [2015/01/29 12:06] – [Example] rupert | lecture_4 [2015/01/29 15:08] (current) – [Discussion] rupert |
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| In both of these solutions we used EROs to transform the [[augmented matrix]] into a nice form. | In both of these solutions we used EROs to transform the [[augmented matrix]] into a nice form. |
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| * In solution 1, we ended up with the matrix $\left[\begin{smallmatrix}1&0&3&0\\0&1&-2&5\\0&0&1&-3\end{smallmatrix}\right]$ which has a staircase pattern, with zeros below the staircase, and 1s just above the "steps" of the staircase. This is an example of a matrix in **row echelon form** (see below). We needed a bit of easy algebra, called **back substitution**, to finish off the solution. | * In solution 1, we ended up with the matrix $\left[\begin{smallmatrix}1&0&3&0\\0&1&-2&5\\0&0&1&-3\end{smallmatrix}\right]$ which has a staircase pattern, with zeros below the staircase, and 1s just above the "steps" of the staircase. This is an example of a matrix in **row echelon form** (see below). We needed a bit of easy algebra, called **back substitution**, to finish off the solution. (Why is it called //echelon// form? It seems that this word has an [[http://dictionary.reference.com/browse/echelon|archaic meaning]] which is relevant to the staircase-like pattern: "any structure or group of structures arranged in a steplike form.") |
| * In solution 2, we ended up with the matrix $\left[\begin{smallmatrix}1&0&0&9\\0&1&0&-1\\0&0&1&-3\end{smallmatrix}\right]$ which has a staircase pattern with zeros below the staircase and 1s just above the "steps" of the staircase, and the additional property that we only have zeros above the 1s on the steps. This is an example of a matrix in **reduced row echelon form** (see below). Finding the solution from this matrix needed no extra algebra. | * In solution 2, we ended up with the matrix $\left[\begin{smallmatrix}1&0&0&9\\0&1&0&-1\\0&0&1&-3\end{smallmatrix}\right]$ which has a staircase pattern with zeros below the staircase and 1s just above the "steps" of the staircase, and the additional property that we only have zeros above the 1s on the steps. This is an example of a matrix in **reduced row echelon form** (see below). Finding the solution from this matrix needed no extra algebra. |
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| \def\ar#1{\\[6pt]\xrightarrow{#1}&} | \def\ar#1{\\[6pt]\xrightarrow{#1}&} |
| &\go{1&2&3&4&5}{0&1&2&3&4}{0&0&1&2&3} | &\go{1&2&3&4&5}{0&1&2&3&4}{0&0&1&2&3} |
| \ar{R1\to R1-3R3\text{ and } R2\to R2-2R3}\go{1&2&0&-2&-4}{0&1&0&-1&-2}{0&0&1&2&3} | \ar{R2\to R2-2R3}\go{1&2&3&4&5}{0&1&0&-1&-2}{0&0&1&2&3} |
| | \ar{R1\to R1-3R3}\go{1&2&0&-2&-4}{0&1&0&-1&-2}{0&0&1&2&3} |
| \ar{R1\to R1-2R2}\go{1&0&0&0&0}{0&1&0&-1&-2}{0&0&1&2&3} | \ar{R1\to R1-2R2}\go{1&0&0&0&0}{0&1&0&-1&-2}{0&0&1&2&3} |
| \end{align*} | \end{align*} |
