1 One Application of Linear Systems: Network...

EK102: Linear Algebra Lectures 5, 6: Feb 6th, 8th 2017 Section C1

Professor Joana Amorim, [email protected]

What is on today

1 One Application of Linear Systems: Network Flow 1

2 Linear Independent sets 3

3 Application of systems to Linear Transformations 5

4 Matrix Operations 74.1 Sum and scalar Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 The inverse of a matrix 115.1 Inverse of 2× 2 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 An Algorithm to find the inverse of n× n matrices . . . . . . . . . . . . . . . . . . . . . . 125.3 Solving systems using the inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Invertibility criteria 14

1 One Application of Linear Systems: Network Flow

In the book:

• Section 1.6, pages 53 – 54.

• Exercises 11 to 14.

• To see more applications have a look at the other two examples in section 1.6, pages 50 – 52 andthe examples in section 1.10.

• Objective: study the flow of some quantity through a network ( traffic flow, current in circuits,distribution of products...)

• Representing a network:

• Arrows represent direction of the flow. If you get negative flow it means things are moving in thedirection opposite to that shown on the model.

• Assumption: Total flow in a junction= Total flow out that junction.

1


Example 1 (Similar to Exercise 11, section 1.6)

2


2 Linear Independent sets

In the book:

• Section 1.7, all.

Definition 2

• A set of vectors {v1, . . . ,vp} in Rn is said to be linearly independent (LI) if the vector equation

x1v1 + x2v2 + . . . + xpvp = 0

has only the trivial solution x = 0.

• The set {v1, . . . ,vp} is said to be linearly dependent (LD) if there is a non-trivial solution (asolution x 6= 0) for the above equation.

Example 3 (Exercise 1, section 1.7) Determine if the vectors are linearly independent. Justify youranswer. 5

00

,

72−6

,

94−8

Example 4 Determine if the vectors are linearly independent. Justify your answer.500

,

72−6

,

194−12

Deciding if a set of vectors is {v1, . . . ,vp} LI or LD IS EQUIVALENT to deciding if the homo-geneous system

x1v1 + x2v2 + . . . + xnvn = 0⇔ [v1 v2 . . . vp]x = 0

has only the trivial solution (LI) or has an infinite number of solutions (LD).

3


Possibilities:

• Sets with 1 vector: if the vector is zero they are LD, if the vector is not 0 the set is LI.

• Sets with two vectors: the set if LD iff one vector is a multiple of the other.

• Sets with two or more vectors: the set is LD iff one vector can be written as a linear combination ofthe rest (ie, one vector is in the span of the other vectors).

Example 5 Decide if the following sets are LI or LD. Justify your answer. 19

4−12

{[24

],

[612

]}{[

24

],

[10

],

[38

]}

Remark 6 • If the zero vector is included in the set, the set is always LD.

• If we have m vectors of Rn, with m > n, they are always LD. (For example 3 or more vectors of R2,4 or more vectors of R3, etc, are always LD).

Example 7 (Exercise 7, section 1.7) Determine if the columns of the matrix below form a LI set.Justify your answer. 1 4 −3 0

−2 −7 5 1−4 −5 7 5

Example 8 (Similar to Exercise 5, section 1.7) Determine if the columns of the matrix below forma LI set. Justify your answer.

0 −3 92 1 −7−1 4 −51 −4 −2

4


Example 9 (Exercise 13, section 1.7) Find the value(s) of h or which the vectors are LD. Justify youranswer. 1

5−3

,

−2−96

,

3h−9

3 Application of systems to Linear Transformations

In the book:

• Parts of Sections 1.8 and 1.9 that involve systems: read example 1 page 65 and then pages 76 –78.

• You can now do all exercises from section 1.8 and 1.9.

Example 10 (Similar to Exercise 11, section 1.8) Let

A =

1 −3 5 −50 1 −3 52 −4 4 −4

,b =

−110

,

and let T : R4 → R3 be defined by T (x) = Ax. Is b in the range of the linear map T? If is the answer isno justify, if the answer is yes find one x whose image under T is b.

5


Definition 11

• A map T : Rn → Rm is said to be onto if each b in Rm is the image of at least one x in Rn (ie,range = codomain, ie T (x) = b has one or infinite solutions).

• A map T : Rn → Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn

(ie, T (x) = b has one or none solutions).

Theorem 12T : Rn → Rm is one-to-one if and only if the equation T (x) = 0 has only the trivial solution (ie, x = 0).

Example 13 (Based on exercise 17, section 1.9) Let T : R4 → R4 be defined by

T (x1, x2, x3, x4) = (x1 + 2x2, 0, 2x2 + x4, x2 − x4).

Is this map one-to-one? Is it onto? Justify your answer.

6


4 Matrix Operations

In the book:

• Section 2.1, pages 94 – 100.

• Exercises 1 – 12.

Notation:

Identity Matrix Zero Matrix Diagonal Matrix Upper Triangular Matrix

4.1 Sum and scalar Multiples

Sum: the sum of two matrices A and B (of the same size!!) is the matrix A + B whose columns are thesums of the corresponding columns in A and B.

Scalar Multiplication: If A is a matrix and c a scalar, the scalar multiple cA is the matrix whose columnsare c time the corresponding columns in A.

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Proposition 14 (Properties of sums and scalar multiples of matrices)

Example 15 (First part of exercise 1, section 2.1) Let

A =

[2 0 −14 −5 −2

]and B =

[7 −5 11 −4 −3

].

Compute each expression if possible. If an expression is undefined explain why.

−2A

B − 2A

4.2 Matrix Multiplication

Matrix multiplication is more sensitive than matrix sum. We can think of it as composition of maps:

So for the product AB to be defined we need to have that

#columns in A = #rows in B.

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If this is the case (A is m× n and B is n× s) we have

(AB)ij = ai1b1j + . . . + ainbnj.

An example illustrates this better:

[2 3 10 1 2

] 1 −23 1−2 6

Proposition 16 (Properties of multiplication of matrices)

Example 17 (Second part of exercise 1, section 2.1) Let

A =

[2 0 −14 5 −2

], C =

[1 2−2 1

]and D =

[3 5−1 4

].

Compute each expression if possible. If an expression is undefined explain why.

AC

CD

9


Example 18 (Similar to Exercise 10, section 2.1) Let

A =

[3 −6−1 2

], B =

[−1 13 4

]and C =

[−3 −52 1

].

Verify that AB = AC and yet B 6= C!

Important remarks about multiplication of matrices

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5 The inverse of a matrix

In the book:

• Section 2.2, pages 104 – 110 except elementary matrices and end of page 108.

• Exercises 1 – 10, 29 – 33.

Idea:

Definition 19 A n × n matrix A is invertible if there exists a matrix C, called the inverse of A, suchthat

AC = CA = In.

If A is invertible, its inverse is unique and represented by A−1.A non-invertible matrix is called singular. An invertible matric is called non-singular.

Proposition 20 (Properties of the Inverse) 1. (A−1)−1 = A.

2. (AB)1 = B−1A−1.

3. (AT )−1 = (A−1)T

Example 21 A =

[2 5−3 −7

], C =

[−7 −53 2

].

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5.1 Inverse of 2× 2 matrices

Determinant:

If A =

[a bc d

]we define detA = determinant of A= ad− bc.

Theorem 22 If detA 6= 0 then A is invertible and

A−1 =1

ad− bc

[d −b−c a

].

If detA = 0 the A is not invertible.

Example 23 (Exercises 1 and 2, section 2.2) Find the inverses of the matrices (if possible).

A =

[8 65 4

]

A =

[3 28 5

]

5.2 An Algorithm to find the inverse of n× n matrices

Theorem 24 An n × n matrix A is invertible if and only if it is row equivalent to In (ie, its reducedechelon form is In). In this case the operations that take A to In also take In to A−1.

Algorithm: Reduce [ A | In ] and you get [ In | A−1 ].

12


Example 25 (Similar to Ex 32, sec 2.2) Find the inverse of A =

1 2 −1−4 −7 3−2 −6 4

.

Example 26 (Similar to Ex 30, sec 2.2) Use the above algorithm to find the inverse of A =

[3 64 7

].

13


5.3 Solving systems using the inverse of a matrix

Theorem 27 Let A be a n× n invertible matrix.Then the system Ax = b has a unique solution x = A−1b.

Example 28 (Exercise 5, section 2.2) Solve the system using the theorem above.{8x1 + 6x2 = 25x1 + 4x2 = −1

6 Invertibility criteria

In the book:

• Section 2.3, pages 113 – 116.

• Exercises all, specially 1 – 8 and 33 – 34.

When can we invert a matrix A? What we already know:

• 2× 2 if detA 6= 0.

• n× n if A −→ In.

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Theorem 29

Example 30 (Exercises 2, 4, 5 and 7 section 2.3) Decide if the following matrices are invertible ornot. Do as little operations as possible.

A =

[−4 26 −3

], B =

−5 1 40 0 01 4 9

, C =

3 0 −32 0 4−4 0 7

15


D =

−1 −3 0 13 5 8 −3−2 −6 3 20 −1 2 1

Remark 31 A linear map T : Rn → Rn is represented by a n× n matrix A:

T is invertible if there exists a map S such that T (S(x)) = x and S(T (x)) = x. S is the inverse of T ,S = T−1.

A map T is invertible iif A is invertible and A−1 is the matrix representation of T−1.

[Do exercises 33 and 34 section 2.3]

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