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MATH1850U/2050U: Chapter 1 1

LINEAR SYSTEMS

Introduction to Systems of Linear Equations (1.1; pg.2)

Definition: A linear equation in the n variables, nxxx ,, 21 is defined to be an

equation that can be written in the form:

bxaxaxa nn 2211 where naaa ,, 21 and b are real constants. The ia are called the coefficients, and the

variables ix are sometimes called theunknowns. If it cannot be written in this form, it is

called a nonlinear equation.

Examples:

4321 81065 xxxx

4

2

231 639 xxxx

97 321 xxx

Application: Suppose that $100 is invested in 3 stocks. If A , B, and C, denote thenumber of shares of each stock that are purchased and they have units costs $5, $1.5, and$3 respectively, write the linear equation describing this scenario.

Definition: A solution of a linear equation bxaxaxa nn 2211 is a sequence

(or n-tuple) ofn numbers nsss ,, 21 such that the equation is satisfied when we

substitute nn sxsxsx ,, 2211 in the equation. The set of ALL solutions of the

equation is called the solution set (or sometimes the general solution) of the equation.

Example:

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Definition: A finite set of linear equations in n variables, nxxx ,, 21 , is called a system

of linear equations or a linear system. A sequence ofn numbers nsss ,, 21 is a

solution of the system of linear equations if nn sxsxsx ,, 2211 is a solution of

every equation in the system.

Exercise: Verify that 1,1,1 zyx is a solution of the linear system:

12

43

232

zy

yx

zyx

Question: Does every system of equations have a solution?

Definition: A system of equations that has no solutions is said to be inconsistent. Ifthere is at least of solution of the system, it is called consistent.

Example (one solution):

Example (no solutions):

Example (infinitely many solutions):

Question: What do the above cases correspond to geometrically?

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Extension: Now lets say we have 3 equations in 3 unknownsthen what? How manysolutions are possible? What is the geometric interpretation?

Ok, finally, lets extend this even furthern equations in n unknowns.

Theorem: A system of linear equations has either no solution, exactly one solution, orinfinitely many solutions.

Ok, so we know what to expect in terms of solutions, but how do we solve so manyequations in so many unknowns???

Definition: An arbitrary set ofm equations in n unknowns

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

2211

22222121

11212111

can be written more concisely in augmented matrix form:

mmnmm

n

n

baaa

baaa

baaa

21

222221

111211

Note: The first row of the matrix corresponds to the coefficients in the 1 st equation, the2nd row to the coefficients of the 2nd equation, etc. Likewise, the 1st column of the matrixcorresponds to the coefficients of the 1

stvariable, the 2

ndcolumn to the coefficients of the

2nd variable, etc.

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Example: Write the linear system in augmented matrix form:

12

43

232

zy

yx

zyx

.

Example: Suppose we have

7

2

3

1

5

8

0

1

2

0

0

1

. What is the solution to the system?

So how does this help???

Example: Use elementary row operations to replace the system of equations withanother system that has the same solution, but is easier to solve. Then solve the resulting

system.

24

03

2

321

321

321

xxx

xxx

xxx

Definition: You find solutions to systems of linear equations using three types of

operations on an augmented matrix (or the system of equations itself, in brackets): Multiply a row (equation) through by a nonzero constant Interchange two rows (equations) Add a multiple of one row (equation) to another

These are called Elementary Row Operations.

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Note: If we continue doing row operations to make the leading entry in each row a 1,then the resulting system is said to be in row echelon form which well discuss in amoment.

Gaussian Elimination (1.2; pg. 11)

Definition: A matrix is said to be in reduced row-echelon form if it has the following 4

properties (if only the 1st

3 properties are satisfied, it is said to be in row-echelon form).1. If a row does not consist entirely of zeroes, then the first nonzero number in the

row is a 1. This is called a leading 1.2. If there are any rows that consist entirely of zeroes, then they are grouped together

at the bottom of the matrix.3. In any two successive rows that do not consist entirely of zeroes, the leading 1 in

the lower row occurs farther to the right than the leading 1 in the higher row.4. Each column that contains a leading 1 has zeroes everywhere else.

Examples: Are these RREF, REF, or neither?

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So, why are we doing this? Well, once youve got a system in one of these forms,finding the solution is super-easy.

Example: Consider the augmented matrices

3161

6114

9523

4112

and

0000

1100

1010

2001

(you cant tell just by looking at them, but the 2nd

matrix and the 1st

one are actuallyequivalent systems...i.e. the 2nd one is the RREF of the 1st one)

More Examples (augmented matrices):

000

710

201

0100

5001

3000

5201

0000

0000

95107801

Terminology: A leading variable is a variable that has a leading 1 in its column and afree variable is a variable that has no leading 1 in its column.

Example:25

1752

32

54321

xx

xxxxx

Now that weve covered some important definitions, lets return to solving the system.

Definition: Solving by reducing to row echelon form and then using back-substitution iscalled Gaussian Elimination. Solving by reducing to reduced-row echelon form iscalled Gauss-Jordan Elimination.

[Pictures from Wikipedia]

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Gaussian Elimination Algorithm (to get matrix to row-reduced form):

1. Locate the 1st non-zero column...if it doesnt contain a leading 1, create a leading1 using an elementary row operation.

2. Move the leading 1 to the top of that column by switching 2 rows.3. Use the leading 1 to create 0s underneath it, using row operations.4. Move onto the next column and repeat steps 1-4 working only with the rows

below the ones that the algorithm has already been applied to.

Example (Gaussian Elimination): Solve the system

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Example (Gauss-Jordan elimination): Solve the system

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More Examples: Lets do some more examples of the end of the algorithm...suppose werow-reduced the augmented matrix corresponding to a linear system and obtained:

Definition: A system of linear equations is said to be homogeneous if the constant termsare all zero. That is, the 0ib for all i between 1 and n.

0

0

0

2211

2222121

1212111

nmnmm

nn

nn

xaxaxa

xaxaxa

xaxaxa

Some Properties of Homogeneous Systems:

Every homogeneous system is consistent, because it has at least the solution0,0,0 21 nxxx . This is called the trivial solution.

Either there is only the trivial solution, or there are infinitely many solutions A homogeneous system of equations with more unknowns than equations has

infinitely many solutions

Example:075

02

321

321

xxx

xxx

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Example: What conditions must 1b , 2b , and 3b satisfy in order for the system of

equations

332

231

1321

2

bxx

bxx

bxxx

to be consistent?

Example: Determine the values ofkfor which the system of equations has:i) no solutions;ii) exactly one solution;iii) infinitely many solutions.

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