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Elementary column operations

All the theory of linear systems we have discussed so far (e.g., matrix form, equivalent systems, elementary row operations, row echelon form, Gaussian elimination) depends on the choice we have initially made of arranging the equations of the system vertically (one below the other) and writing their left- and right-hand sides as entries of two column vectors.

A parallel, and mostly identical, theory can be developed by arranging the equations horizontally (one besides the other) and writing their left- and right-hand sides as entries of two row vectors. In this parallel theory, which is briefly outlined below, elementary column operations (which are the counterpart of elementary row operations) play a key role. Furthermore, precisely defining column operations is worthwhile because several useful concepts in linear algebra rely on a unified treatment of row and column operations.

Table of Contents

Systems in row form

Remember that a system of K linear equations in $L$ unknowns [eq1]can be represented in matrix form as[eq2]where A is the $K	imes L$ matrix of coefficients, $b$ is the Kx1 vector of constants and x is the $L	imes 1$ vector of unknowns.

Thus, we have two Kx1 column vectors on the two sides of the equal sign:

By transposing both sides of the system, we can equivalently write it as[eq3]so that the vector of unknowns $x^{	op }$ is a $1	imes L$ row vector, the matrix of coefficients $A^{	op }$ is a $L	imes K$ matrix and the vector of constants $b^{	op }$ is a $1	imes K$ row vector.

In other words, we have two $1	imes K$ row vectors on the two sides of the equal sign:

This is what we meant in the introduction by "arranging the equations horizontally, one besides the other".

Notice that:

Example Consider the system[eq5]Then, [eq6]and we have two $2	imes 1$ vectors on the two sides of the equal sign:[eq7]Each row of the two vectors corresponds to an equation. By transposing everything, we get[eq8]and we have two $1	imes 2$ vectors on the two sides of the equal sign:[eq9]Each column of the two vectors corresponds to an equation. To better visualize the correspondence between columns, we could write[eq10]

Column operations

When a system is written horizontally, we can obtain systems equivalent to it by performing elementary column operations:

  1. multiplying a column by a non-zero constant;

  2. adding a multiple of one column to another column;

  3. interchanging columns.

These operations are completely analogous to the elementary row operations performed on systems written vertically.

Remember that elementary row operations can be performed in two alternative ways:

  1. directly on the rows of the system;

  2. on the rows of the identity matrix; the system is then pre-multiplied by the resultant matrix.

The same is true of elementary column operations, who can be performed:

  1. directly on the columns of the system;

  2. on the columns of the identity matrix; the system is then post-multiplied by the resultant matrix.

This can be easily seen as follows. When the system is arranged vertically and the matrix obtained by performing elementary row operations is $R$, then the equivalent system is[eq11]

If we transpose both sides of the equation, we get[eq12]where $R^{	op }$ is the matrix obtained by performing column operations on the identity matrix (transposition transforms row operations into column operations).

Example Consider the system of two equations in three unknowns[eq13]that can be written in matrix form as [eq14]where [eq15]Adding the first equation to the second, we obtain the equivalent system[eq16]that can be written in matrix form as[eq17]where[eq18]We obtain the same result by 1) taking the $2	imes 2$ identity matrix[eq19]2) adding its first column to the second:[eq20]and 3) post-multiplying $A^{	op }$ and $b^{	op }$ by $R^{	op }$:[eq21]

Column echelon form

Remember that systems arranged vertically are easy to solve when they are in row echelon form or reduced row echelon form. These forms have obvious counterparts for systems arranged horizontally: the system[eq14]is said to be in (reduced) column echelon form if and only if the system [eq2]is in (reduced) row echelon form.

Equivalently, the matrix $A^{	op }$ is said to be in (reduced) column echelon form if and only if the matrix A is in (reduced) row echelon form.

Gaussian and Gauss Jordan elimination

Finally, Gaussian and Gauss Jordan elimination, the two algorithms used to transform a vertical system into an equivalent system in (reduced) row echelon form, can be used on a horizontal system with straightforward modifications: whenever an elementary row operation is necessary for the vertical system, we instead perform a column operation on the horizontal system.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Write the system[eq24]horizontally.

Solution

The system is[eq25]

Exercise 2

Write the matrix $R$ that allows to sum twice the first column of the system in the previous exercise to the second column.

Solution

The matrix $R$ is obtained by performing the elementary column operation on the $2	imes 2$ identity matrix: [eq26]

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