# Linear Equations in Linear Algebra

** Operations on a System of Linear Equations **

Operations on a system of linear equations which do not
change its set of solutions:

(1) multiply an equation by a scalar (number)

(2) replace an equation by the sum of itself and a scalar multiple of another
equation

(3) interchange two equations

**Elementary Row Operations on a Matrix**

A system of linear equations can be represented by a
matrix. Operations on the equations correspond to

elementary row operations on its matrix.

(1) multiply a row by a scalar ( number )

(2) replace a row by the sum of itself and a scalar multiple of another row

(3) interchange two rows

**Definitions**

(a) pivot

First non-zero entry in a row.

(b) echelon form

Each pivot occurs to the left of all pivots below it.

Any zero row occurs below all non-zero rows.

(c) reduced echelon form

Each pivot is 1.

Each pivot is the only non-zero entry in its column.

**Algorithms for Solving a System of Linear Equations**

(a) **Gauss-Jordan**

Use Gaussian elimination to arrive at the reduced echelon form of the matrix.

(b) **Gaussian Elimination with Back- Substitution **

Use Gaussian elimination to arrive at an echelon form of the matrix, then use
back-substitution.

**Case Study: Linear Models in Economics**

Download the supporting pdf file and Mathematica notebook from the Lay Linear Algebra web site.

**An Algebraic Point of View**

Consider the set of
all systems of linear equations with real coefficients having m equations and n

unknowns . Declare one such system to be equivalent to another if there is a
sequence of elementary operations

(scale, swap, or replace) which transform the first into the second. This
relationship is an equivalence relation,

and the solution set of such a linear system of equations is an invariant of its
equivalence class. This is just

another way of saying that any two systems of linear equations which are
equivalent to each other have the

same solution set.

Now consider the set of
m by n+1 matrices with real coefficients . Declare one such matrix to

be equivalent to another if there is a sequence of elementary row operations
which transforms the first into

the second. This defines an equivalence relation on this set of matrices, viewed
as augmented matrices of

linear systems, so that these matrices break up into equivalence classes.

The map which sends a
system of linear equations to its augmented matrix

is a bijection, and it respects equivalence classes. Thus,
implies that

, where the first
is an equivalence of linear
systems and the second is an equivalence of

augmented matrices. It works the other way as well, since elementary operations
are invertible and their

inverses are other elementary operations. Thus, if two augmented matrices are
equivalent, then the linear

systems they represent are also equivalent, and therefore they have the same
solution sets.

This is the fundamental fact behind our interest in
echelon forms and the like . They classify matrices, and

hence classify linear systems, in a manner which is invariant over equivalence
classes, hence invariant over

solution sets. The following example illustrates this sort of relationship.

Suppose that a certain matrix A is equivalent to the
reduced echelon matrix U, and it is clear by inspection

that the third column of U is a linear combination of its two previous columns.
Then the third column of

A is also a linear combination of its two previous columns, because the related
linear systems have the same

solutions. In fact, the coefficients which express this linear dependence are
the same since

For a specific example of this sort of thing, consider that

It is clear that the third column of the matrix on the
right is a linear combination of the previous two columns.

Indeed, as well, since the vector x = (−1,
2,−1) is a

solution of both associated linear systems: Ax = 0 and Ux = 0.

**Exercises**

We will solve some of the following exercises as a
community project in class today. Finish these solutions

as homework exercises, write them up carefully and clearly, and hand them in at
the beginning of class next

Friday. You are encouraged to use a computer algebra system whenever
appropriate.

Exercises for Lay, Section 1.1, pp 11–13: 1, 3, 5, 9, 13,
15, 33, 34 (heat transfer)

Exercises for Lay, Section 1.2, pp 25–27: 1, 3, 7, 11, 13, 17, 19, 33
( interpolating polynomial ),

34 (wind tunnel experiment)

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