Maths linear algebra

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Singapore University of

Technology & Design

MATH 10.004Systems of Linear Equations

Cohort 1

Meyer, Sections 1.1-1.2


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Linear Equations Singularity and Non-Singularity Summary

LEARNING OBJECTIVES

After this cohort you will be able to ...

be able to draw and think about planes in multiple dimensions

based on a linear equation.

understand the geometry of systems of linear equations in terms

of planes.

categorize a system of equations by the existence and number ofits solutions.

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ACTIVITY 1: LINEAR EQUATIONS

Consider the equation: 2x`4y3z5.

1. Find a solution of this equation.

2. Subtract 2 from thexvalue of your solution and add 1 to the y

value. Is it still a solution?

3. Why is there more than one solution to this equation?

4. Find all solutions to this equation.

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ACTIVITY 1: LINEAR EQUATIONS

Consider the equation: 2x`4y3z5.

1. Find a solution of this equation.

px, y, zq p2, 1, 1qis a solution.

2. Subtract 2 from thexvalue of your solution and add 1 to the y

value. Is it still a solution? Why?

px, y, zq p0, 2, 1qis still a solution because2p2q `4p1q 3p0q 0.

3. Why is there more than one solution to this equation?

There are 3 unknownspx, y, zqand only one equation*.

4. Find all solutions to this equation.

If we letyP R andzP R, thenx 12

p54`3q.

Thus all solutions are described by:`1

2 p54`3q, , .4


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SYSTEM OF LINEAR EQUATIONS

Alinear systemof mequations innunknowns is written as

a11x1à12x2` a1nxnb1

a21x1à22x2` a2nxnb2...

am1x1àm2x2` amnxnbm.

which is a mathematical way of expressingmlinear equalityconstraints that thenvariablesxi, iP t1, . . . , nu, need to satisfy.

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GEOMETRIC INTERPRETATION

a11x1à12x2` a1nxnb1

a21x1à22x2` a2nxnb2...

am1x1àm2x2` amnxnbm.

mhyperplanes inndimensions, each of the form

ai1x1ài2x2` ainxnbi.

In R2 this is a line. In R3 this is a plane. A system of linear

equations representsmsimultaneous expressions of this form.

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Li E i Si l i d N Si l i S


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SOLUTIONS IN R2

a11x1à12x2b1 is a line

a21x1à22x2b2 is a line

a11x1à12x2b1

x2

a11

a12 x1`

b1

a12

x1

x2

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Li E ti Si l it d N Si l it S


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SOLUTIONS IN R2

The system can have either:

x1

x2

One Solution

x1

x2

No Solutions

x1

x2

Infinite Solutions

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Linear Equations Singularity and Non Singularity Summary


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ACTIVITY 2: GEOMETRIC INTERPRETATION (10 MINUTES)

Consider the linear system:

2xy1

x`y5

Draw the linear system in R2 using the geometric interpretation of

hyperplanes.

Solve forpx, yqand indicate how the solution appears in your figure.

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ACTIVITY 2: GEOMETRIC INTERPRETATION

x

y 2xy1

x`y5

px, yq p2, 3q

Theuniquesolution to

"2xy 1x`y 5

* is given by

px, yq p2, 3q, the point where the two lines cross.

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SINGULARITY AND NON-SINGULARITY

A system of linear equations can be either: Consistent

Unique solution (Non-Singular) Infinite number of solutions (Singular)

Inconsistent No solution (Singular)

A system of linear equations can be:

Underdetermined(underconstrained) more unknowns (variables) than equations (constraints)

Overdetermined(overconstrained)

more equations (constraints) than unknowns (variables)

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ACTIVITY 3: SOLUTION GEOMETRY (5 MINUTES)

The following plots show three planes in R3, where each plane

corresponds to one of the equations in a system of 3 linear equations

in 3 unknowns.

For each plot, specify whether the system described is consistentor

inconsistent, and if consistent specify the number of solutions.

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q g y g y y

ACTIVITY 3: SOLUTION GEOMETRY

For each plot, specify whether the system described is consistentor

inconsistent, and if consistent specify the number of solutions.

Consistent? # Solutions? Singular?

1 Inconsistent None Singular

2 Consistent Infinite Singular



5 Consistent Unique Nonsingular

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q g y g y y

ACTIVITY 4: PLANES IN R4 (15 MINUTES)

Consider the following three hyperplanes in R4:

u`v`w`z6

u`w`z4

u`w2

1. Describe the intersection of the hyperplanes.

2. Is the intersection a line, a point, or the empty set?

3. What is the intersection if a 4th plane,u 1, is included?

4. Find a 4th equation, which results in no solution.

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ACTIVITY 4: PLANES IN R4

u`v`w`z6 (1)u`w`z4 (2)

u`w2 (3)

1. Describe the intersection of the hyperplanes.(2)-(3) yieldsz2.(1)-(2) yieldsv2.

Hence, our system simplifies to the equivalent systemvz2

andu`w2.

2. Is the intersection a line, a point, or an empty set?

The coordinates ofvandzare fixed anduandware related via

u`w2. This describes a line.

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ACTIVITY 4: PLANES IN R4

u`v`w`z6u`w`z4

u`w2

3. What is the intersection if a 4th plane,u 1, is included?

With this additional constraint we have vz2 (as before) andu 1 andw3. Hence, the intersection now describes apoint.

4. Find a 4th equation, which results in no solution.

u`w5 would be such an equation since it contradictsu`w2.

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ACTIVITY 5: SINGULARITY IN R3 (10 MINUTES)

Consider the following system in R3:

u` v` w2

u`2v`3w1

v`2w0

Explain why the system is inconsistent by finding a combination of thethree equations that adds to 0 1.

Replace the zero on the RHS to allow the equations to have solutions,

and what is one of the solutions?

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ACTIVITY 5: SINGULARITY IN R3

Consider the following system in R3:

u` v` w2 (1)

u`2v`3w1 (2)

v`2w0 (3)

Explain why the system is inconsistent by finding a combination of thethree equations that adds to 0 1.

Replace the zero on the RHS to allow the equations to have solutions,

and what is one of the solutions?

Combining (1)-(2)+(3) yields 0 1. Thus the system is inconsistentand singular.

Consistency (hence a solution) is achieved by setting the zero on the

RHS to1. In this case, one solution is u3,v 1, andw0.

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SUMMARY

The geometric interpretation of linear equations corresponds to

intersecting hyperplanes.

A linear system can have no solutions, a unique solution, or an

infinite number of solutions.

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Maths linear algebra