2.8 Let A = [A1. A2. Aal be a 3 × 3 matrix with linearly independent columns Ai.
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Question
2.8 Let A = [A1. A2. Aal be a 3 × 3 matrix with linearly independent columns Ai. (a) Explain why the row reduced form of A is the following matrix R EXERCISES 111 Hint: Think about the number of free variables in the system with augmented matrix [A, 0].] R=|0 | 0 0 0 1 (b) Let A = [Al,A2,A3. A4Ag] be a 3x5 matrix such that the first three columns are linearly independent. Explain why the pivot columns must be the first three. [Hint: If not, what would this say about the row reduced form of [Aj.A2. A3]?]Explanation / Answer
A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
The reduced row echelon form of a matrix may be computed by Gauss–Jordan elimination. Unlike the row echelon form, the reduced row echelon form of a matrix is unique and does not depend on the algorithm used to compute it. For a given matrix, despite the row echelon form not being unique, all row echelon forms and the reduced row echelon form have the same number of zero rows and the pivots are located in the same indices.
This is an example of a matrix in reduced row echelon form:
{displaystyle left[{egin{array}{ccccc}1&0&a_{1}&0&b_{1}\0&1&a_{2}&0&b_{2}\0&0&0&1&b_{3}end{array}} ight]}
Note that this does not always mean that the left of the matrix will be an identity matrix, as this example shows.
For matrices with integer coefficients, the Hermite normal form is a row echelon form that may be calculated using Euclidean division and without introducing any rational number or denominator. On the other hand, the reduced echelon form of a matrix with integer coefficients generally contains non-integer coefficients.
Transformation to row echelon form.
By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to row echelon form. Since elementary row operations preserve the row space of the matrix, the row space of the row echelon form is the same as that of the original matrix.
The resulting echelon form is not unique; any matrix that is in echelon form can be put in an (equivalent) echelon form by adding a scalar multiple of a row to one of the above rows, for example:
{displaystyle {egin{bmatrix}1&3&-1\0&1&7\end{bmatrix}}{xrightarrow { ext{add row 2 to row 1}}}{egin{bmatrix}1&4&6\0&1&7\end{bmatrix}}.}
However, every matrix has a unique reduced row echelon form. In the above example, the reduced row echelon form can be found as
{displaystyle {egin{bmatrix}1&3&-1\0&1&7\end{bmatrix}}{xrightarrow { ext{subtract 3 times row 2 from row 1}}}{egin{bmatrix}1&0&-22\0&1&7\end{bmatrix}}.}
This means that the nonzero rows of the reduced row echelon form are the unique reduced row echelon generating set for the row space of the original matrix.
Systems of linear equations
A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of equations is said to be in reduced row echelon form or in canonical form if its augmented matrix is in reduced row echelon form.
The canonical form may be viewed as an explicit solution of the linear system. In fact, the system is inconsistent, if and only if one of the equations of the canonical form is reduced to 0 = 1. Otherwise, regrouping in the right hand side all the terms of the equations but the leading ones, expresses the variables corresponding to the pivots as constants or linear functions of the other variables, if any
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