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10212 Linear Algebra B


University of Manchester

27 January 2020


Textbook

Students are strongly advised to acquire a copy of the Textbook:

D. C. Lay. Linear Algebra and its Applications. Pearson, 2006.ISBN 0-521-28713-4. (Or other editions)

Lecture notes serve only as indication of the course content.


Homework:

I Homework x has to be returned for marking before 09:00 onFriday in Week x − 1.

I consists of some odd numbered exercises from the Textbook.

I Textbook contains answers to most odd numbered exercises.


Communication

I Course website:https://personalpages.manchester.ac.uk/staff/alexandre.borovik/math10212.html

or the same location with the shortened address:https://bit.ly/2Ba17GU

I Email: Feel free to send questions, etc., [email protected]

but only from your university’s e-mail account.Emails from Gmail, Hotmail, etc. automatically go to spam.


What linear Algebra is about?

Linear Forms

A The total cost of a purchase of amounts g1, g2, g3 of somegoods at unit prices p1, p2, p3 is

p1g1 + p2g2 + p3g3 =3∑

i=1

pigi .

Expressions of this kind,

a1x1 + · · ·+ anxn

are called linear forms in variables x1, . . . , xn with coefficientsa1, . . . , an.



Linear Algebra is mathematics of linear forms

B

I Over the course, we shall develop increasingly compactnotation for operations of Linear Algebra.In particular,

p1g1 + p2g2 + p3g3

can be very conveniently written as

[p1 p2 p3

] g1g2g3



Compression of notation

C

I . . . and then abbreviated

[p1 p2 p3

] g1g2g3

= PTG ,

where

P =

p1p2p3

and G =

g1g2g3



Linear Algebra for Physicists

D

I Physicists use even shorter notation and, instead of

p1g1 + p2g2 + p3g3 =3∑

i=1

pigi

writep1g

1 + p2g2 + p3g

3 = pigi .

This notation was introduced by Albert Einstein.Will not be used in the course.



Warning:

I Increasingly compact notation leads to increasingly compactand abstract language used.

I Linear Algebra focuses on the development of a specialmathematics language rather than on procedures.

I This language is used all over mathematics and statistics.



Prerequisites:

More abstract bits of MATH10111:

I Functions: 1–1, onto,bijective.

I Equivalence relations.

I Binary operations and groups.


Lecture 1: Systems of linear equations

Linear equation

A A linear equation in the variables x1, . . . , xn is an equationthat can be written in the form

a1x1 + a2x2 + · · ·+ anxn = b

where b and the coefficients a1, . . . , an are real numbers. Thesubscript n can be any natural number.



B A system of linear equations is a collection of one or morelinear equations involving the same variables, say x1, . . . , xn. Forexample,

x1 + x2 = 3

x1 − x2 = 1



C A solution of the system is a list (s1, . . . , sn) of numbers thatmakes each equation a true identity when the values s1, . . . , sn aresubstituted for x1, . . . , xn, respectively.

The set of all possible solutions is called the solution set of thelinear system.

Two linear systems are equivalent if they have the same solutionset.


Lecture 2: Elementary operations

A A solution of the system is a list (s1, . . . , sn) of numbers thatmakes each equation a true identity when the values s1, . . . , sn aresubstituted for x1, . . . , xn, respectively.

The set of all possible solutions is called the solution set of thelinear system.

Two linear systems are equivalent if the have the same solutionset.



B We shall be use the following elementary operations onsystems od simultaneous liner equations:

Replacement Replace one equation by the sum of itself and amultiple of another equation.

Interchange Interchange two equations.

Scaling Multiply all terms in a equation by a nonzeroconstant.



C Note: The elementary operations are reversible.

D Theorem: Elementary operations preserve equivalence.

If a system of simultaneous linear equations is obtained fromanother system by elementary operations, then the two systemshave the same solution set.



E A system of linear equations has either

I no solution, or

I exactly one solution, or

I infinitely many solutions.

A system of linear equations is said to be consistent it if hassolutions (either one or infinitely many), and a system ininconsistent if it has no solution.



F A system of linear equations has either

I no solution, or

I exactly one solution, or

I infinitely many solutions.

A system of linear equations is said to be consistent it if hassolutions (either one or infinitely many), and a system ininconsistent if it has no solution.



Matrix notation

G The system

x1 − 2x2 + 3x3 = 1

x1 + x2 = 2

x2 + x3 = 3

has the matrix of coefficients1 −2 31 1 00 1 1



H . . . and the augmented matrix1 −2 3 11 1 0 20 1 1 3

;

notice how the coefficients are aligned in columns, and howmissing coefficients are replaced by 0.



I A matrix with m rows and n columns is called an m× nmatrix.



Elementary row operations

J

Replacement Replace one row by the sum of itself and a multipleof another row.

Interchange Interchange two rows.

Scaling Multiply all entries in a row by a nonzero constant.

The two matrices are row equivalent if there is a sequence ofelementary row operations that transforms one matrix into theother.



K Note: the row operations are reversible.

If the augmented matrices of two linear systems are rowequivalent, then the two systems have the same solutionset.


Lecture 3: Row reduction and Echelon form

Existence and uniqueness questions

A

I Is the system consistent?

I If a solution exist, is it unique?



Leading entries

B A nonzero row or column of a matrix is a row or columnwhich contains at least one nonzero entry.

C A leading entry of a row is the leftmost nonzero entry (in anon-zero row).



Echelon form

D A matrix is in echelon form (or row echelon form) if it hasthe following three properties:

1. All nonzero rows are above any row of zeroes.

2. Each leading entry of a row is in column to the right of theleading entry of the row above it.

3. All entries in a column below a leading entry are zeroes.



Reduced echelon form

E If, in addition, the following two conditions are satisfied,

4. All leading entries are equal 1.

5. Each leading 1 is the only non-zero entry in its column

then the matrix is in reduced echelon form.



Row reduction

F An echelon matrix is a matrix in echelon form.

Any non-zero matrix can be row reduced (that, transformed byelementary row transformations) into a matrix in echelon form (butthe same matrix can give rise to different echelon forms).



Examples

G The following is a schematic presentation of an echelonmatrix: � ∗ ∗ ∗ ∗

0 � ∗ ∗ ∗0 0 0 � ∗



H and this is a reduced echelon matrix:1 0 ∗ 0 ∗0 1 ∗ 0 ∗0 0 0 1 ∗



Uniqueness of the reduced echelon form

I

Theorem: Uniqueness of the reduced echelon form.

Each matrix is row equivalent to one and only one reduced echelonform.



Row equivalence is an equivalence relation on the set on m × nmatrices: it is

I reflexive

I symmetric

I transitive

Every equivalence class contains exactly one matrix in reducedechelon form.


Lecture 4: Solution of Linear Systems

Pivot positions

A A pivot position in a matrix A is a location in A thatcorresponds to a leading 1 in the reduced echelon form of A. Apivot column is a column of A that contains a pivot position.



The Row Reduction Algorithm

B 0 2 2 2 21 1 1 1 11 1 1 3 31 1 1 2 2

A pivot is a nonzero number in a pivot position which is used tocreate zeroes in the column below it.



A rule for row reduction:

C

1. Pick the leftmost non-zero column; interchange rows, ifneeded, to make its topmost entry non-zero; it is a pivot.

2. Using scaling, make the pivot equal 1.

3. Using replacement row operations, kill all non-zero entries inthe column below the pivot.

4. Mark the row and column containing the pivot as pivoted.

5. Repeat the same with the matrix made of not pivoted yetrows and columns.

6. Using replacement row operations, kill all non-zero entries inthe column above the pivot entries.



Solution of Linear Systems

DWhen we converted the augmented matrix of a linear system intoits reduced row echelon form, we can write out the entire solutionset of the system.



E Let 1 0 −5 10 1 1 40 0 0 0

be the augmented matrix of a a linear system; then the system isequivalent to

x1 − 5x3 = 1

x2 + x3 = 4

0 = 0



F The variables x1 and x2 correspond to pivot columns in thematrix and a re called basic variables (also leading or pivotvariables).

G The other variable, x3 is a free variable.

Free variables can be assigned arbitrary values and then leadingvariables expressed in terms of free variables:

x1 = 1 + 5x3

x2 = 4− x3

x3 is free



Theorem: Existence and Uniqueness

H A linear system is consistent if and only if the rightmostcolumn of the augmented matrix is not a pivot column—that is, ifand only if an echelon form of the augmented matrix has no row ofthe form [

0 · · · 0 b]

with b nonzero

I If a linear system is consistent, then the solution set containseither

(i) a unique solution, when there are no free variables, or

(ii) infinitely many solutions, when there is at least one freevariable.


Lecture 5: Vector equations

Vectors

A A matrix with only one column is called a column vector, orsimply a vector.

Two vectors are equal if and only if they have the same number ofrows and their corresponding entries are equal.

The set of al vectors with n entries is denoted Rn.



Vectors

B A matrix with only one column is called a column vector, orsimply a vector.





Vectors

C A matrix with only one column is called a column vector, orsimply a vector.





Operations on vectorsD

The sum u + v of two vectors u and v in Rn is obtained by addingcorresponding entries in u and v.[

12

]+

[−1−1

]=

[01

].

The scalar multiple cv of a vector v and a real number (“scalar”)c is the vector obtained by multiplying each entry in v by c .

1.5

10−2

=

1.50−3

.The vector of all zeroes is called the zero vector and denoted 0:

0 =

0...0

.



Operations on vectorsE


12

]+

[−1−1

]=

[01

].


1.5

10−2

=

1.50−3

.

The vector of all zeroes is called the zero vector and denoted 0:

0 =

0...0

.



Operations on vectorsF


12

]+

[−1−1

]=

[01

].


1.5

10−2

=

1.50−3

.The vector of all zeroes is called the zero vector and denoted 0:

0 =

0...0

.



Algebraic properties of Rn

GFor all u, v,w ∈ Rn and all scalars c and d :

1. u + v = v + u

2. (u + v) + w = u + (v + w)

3. u + 0 = 0 + u = u

4. u + (−u) = −u + u = 0

5. c(u + v) = cu + cv

6. (c + d)u = cu + du

7. c(du) = (cd)u

8. 1u = u

(Here −u denotes (−1)u.)



Linear combinations

HGiven vectors v1, v2, . . . , vp in Rn and scalars c1, c2, . . . , cp, thevector

y = c1v1 + · · · cpvp

is called a linear combination of v1, v2, . . . , vp with weightsc1, c2, . . . , cp.



Rewriting a linear system as a vector equationI

x2 + x3 = 2

x1 + x2 + x3 = 3

x1 + x2 − x3 = 2

can be written as equality of two vectors: x2 + x3x1 + x2 + x3x1 + x2 − x3

=

232

which is the same as

x1

011

+ x2

111

+ x3

11−1

=

232



Vector equation

J Denote

a1 =

011

, a2 =

111

, a3 =

11−1

and

b =

232

,then the vector equation can be written as

x1a1 + x2a2 + x3a3 = b.



Solution set of a vector equation

K Solving a linear system is the same as finding an expressionof the vector of the right part of the system as a linearcombination of columns in its matrix of coefficients.



Example

L Write the matrix 0 1 1 21 1 1 31 1 −1 2

in a way that calls attention to its columns:[

a1 a2 a3 b]



Solution set of a vector equation

M A vector equation

x1a1 + x2a2 + · · ·+ xnan = b.

has the same solution set as the linear system whose augmentedmatrix is [

a1 a2 · · · a3 b]

In particular b can be expressed by a linear combination ofa1, . . . , an if and only if there is a solution of the correspondinglinear system.



Span

N If v1, . . . , vp are in Rn, then the set of all linear combinationof v1, . . . , vp is denoted by Span{v1, . . . , vp} and is called thesubset of Rn spanned (or generated) by v1, . . . , vp.

That is, Span{v1, . . . , vp} is the collection of all vectors which canbe written in the form

c1v1 + c2v2 + · · ·+ cpvp

with c1, . . . , cp scalars.



Span

O

Span{v1, . . . , vp} = {b : x1v1 + · · · xpvp = b

has a solution }


Lecture 6: The matrix equation Ax = b

Matrix-vector product

A is m × n matrix, with columns a1, . . . , an

x is in Rn

A The product of A and x, denoted Ax, is the linearcombination of the columns of A using the corresponding entries inx as weights:

Ax =[a1 a2 · · · an

] x1...xn

= x1a1 + x2a2 + · · ·+ xnan



ExampleB The system

x2 + x3 = 2

x1 + x2 + x3 = 3

x1 + x2 − x3 = 2

was written asx1a1 + x2a2 + x3a3 = b.

where

a1 =

011

, a2 =

111

, a3 =

11−1

and

b =

232

.



The same system in the matrix product notation

C 0 1 11 1 11 1 −1

x1x2x3

=

232

or

Ax = b

where

A =[a1 a2 a3

], x =

x1x2x3

.



Solution set of a matrix equation

D Theorem. If A is an m × n matrix, with columns a1, . . . , an,and if x is in Rn, the matrix equation

Ax = b

has the same solution set as the vector equation

x1a1 + x2a2 + · · ·+ xnan = b

which has the same solution set as the system of linear equationswhose augmented matrix is[

a1 a2 · · · an b]

=[A|b].



Existence of solutions

E The equation Ax = b has a solution if and only if b is a linearcombination of columns of A.

F Theorem. Let A be an m × n matrix. Then the followingstatements are equivalent.

(a) For each b ∈ Rn, the equation Ax = b has a solution.

(b) Each b ∈ Rn is a linear combination of columns of A.

(c) The columns of A span Rn.

(d) A has a pivot position in every row.



Row-vector rule for computing Ax

GIf the product Ax is defined then the ith entry in Ax is the sum ofproducts of corresponding entries from the row i of A and from thevector x.



Properties of the matrix-vector product

HTheorem.

If A is an m × n matrix, u, v ∈ Rn, and c is a scalar, then

(a) A(u + v) = Au + Av;

(b) A(cu) = c(Au).



Homogeneous linear systems

IA linear system is homogeneous if it can be written as

Ax = 0.

A homogeneous system always has at least one solution x = 0(trivial solution).

JTherefore for homogeneous systems an important question osexistence of a nontrivial solution, that is, a nonzero vector x whichsatisfies Ax = 0:

The homogeneous system Ax = b has a nontrivial solution if andonly if the system has at least one free variable.



Homogeneous linear systems

KA linear system is homogeneous if it can be written as

Ax = 0.

A homogeneous system always has at least one solution x = 0(trivial solution).

LTherefore for homogeneous systems an important question osexistence of a nontrivial solution, that is, a nonzero vector x whichsatisfies Ax = 0:

The homogeneous system Ax = b has a nontrivial solution if andonly if the system has at least one free variable.



Example

M

x1 + 2x2 − x3 = 0

x1 + 3x3 + x3 = 0


Lecture 7: Linear independence

Nonhomogeneous systemsWhen a nonhomogeneous system has many solutions, the generalsolution can be written in parametric vector form a one vector plusan arbitrary linear combination of vectors that satisfy thecorresponding homogeneous system.

A

Example.x1 + x2 + x3 = 1.

B

Example.

x1 + 2x2 − x3 = 0

x1 + 3x3 + x3 = 5



Solution of nonhomogeneous system

CTheorem. Suppose the equation

Ax = b

is consistent for some given b, and p be a solution. Then thesolution set of Ax = b is the set of all vectors of the form

w = p + vh,

where vh is any solution of the homogeneous equation

Ax = 0.



Linear independence

DAn indexed set of vectors

{ v1, . . . , vp }

in Rn is linearly independent if the vector equation

x1v1 + · · ·+ xpvp = 0

has only trivial solution.



Linear dependence

EThe set

{ v1, . . . , vp }

in Rn is linearly dependent if there exist weights c1, . . . , cp, notall zero, such that

c1v1 + · · ·+ cpvp = 0



Linear independence of matrix columns

FThe matrix equation

Ax = 0

where A is made of columns

A =[a1 · · · an

]can be written as

x1a1 + · · ·+ xnan = 0



Linear independence of matrix columns

GThe columns of matrix A are linearly independent iff the equation

Ax = 0

has only the trivial solution.



HA set of one vectors { v1} is linearly dependent if v1 = 0.

A set of two vectors { v1, v2 } is linearly dependent if at least oneof the vectors is a multiple of the other.



Theorem: Characterisation of linearly dependent sets

IAn indexed set

S = { v1, . . . , vp }

of two or more vectors is linearly dependent if and only if at leastone of the vectors in S is a linear combination of the others.



Theorem: dependence of “big” sets

JIf a set contains more vectors than entries in each vector, then theset is linearly dependent.

Thus, any set{ v1, . . . , vp }

in Rn is linearly dependent if p > n.


Lecture 8: Introduction to Linear Transformations

Transformation

AA transformation T from Rn to Rm is a rule that assigns to eachvector x in Rn a vector T (x) in Rm.

Rn is the domain of T .

Rm is the codomain of T .



Matrix transformations

B

T : Rn −→ Rm

x 7→ Ax

where A is an m × n matrix.

In short:T (x) = Ax.



The range of a matrix transformation

CThe range of T is the set of all linear combinations of the columnsof A.

Indeed, each image T (x) has the form

T (x) = Ax = x1a1 + · · ·+ xnan



Linear transformations

DA transformation T : Rn −→ Rm is linear if:

I T (u + v) = T (u) + T (v) for u, v ∈ Rn;

I T (cu) = cT (u) for all u and all scalars c .



Properties of linear transformations

EIf T is a linear transformation then

T (0) = 0

andT (cu + dv) = cT (u) + dT (v).



The identity matrix

FAn n × n matrix with 1’s on the diagonal and 0’s elsewhere iscalled the identity matrix In:

[1],

[1 00 1

],

1 0 00 1 00 0 1

,

1 0 0 00 1 0 00 0 1 00 0 0 1



The identity transformation

GIt is easy to check that

Inx = x for all x ∈ Rn

Therefore the linear transformation associated with the identitymatrix is the identity transformation of Rn:

Rn −→ Rn

x 7→ x



The matrix of a linear transformation

H Let T : Rn −→ Rm be a linear transformation.Then there exists a unique matrix A such that

T (x) = Ax for all x ∈ Rn.

In fact, A is the m × n matrix whose jth column is the vectorT (ej) where ej is the jth column of the identity matrix in Rn:

A =[T (e1) · · · T (en)

]

A is the standard matrix for the linear transformation T .



The matrix of a linear transformation

I Let T : Rn −→ Rm be a linear transformation.Then there exists a unique matrix A such that

T (x) = Ax for all x ∈ Rn.

In fact, A is the m × n matrix whose jth column is the vectorT (ej) where ej is the jth column of the identity matrix in Rn:

A =[T (e1) · · · T (en)

]A is the standard matrix for the linear transformation T .



Onto and one-to-one

JA transformation T : Rn −→ Rm is onto Rm if each b ∈ Rm is theimage of at least one x ∈ Rn.

T is one-to-one if each b ∈ Rm is the image of at most onex ∈ Rn.



One-to-one: a criterion

KA linear transformation T : Rn −→ Rm is one-to-one

iff

the equation T (x) = 0 has only the trivial solution.



One-to-one and onto in terms of matrices

LLet T : Rn −→ Rm be linear transformation and let A be thestandard matrix for T . Then:

I T maps Rn onto Rm if and only if the columns of A span Rm.

I T is one-to-one if and only if the columns of A are linearlyindependent.


Lecture 9: Matrix operations

Labeling of matrix entries

ALet A be an m × n matrix.

A =[a1 a2 · · · an

]

=

a11 · · · a1j · · · a1n

......

...ai1 · · · aij · · · ain...

......

am1 · · · amj · · · amn



Diagonal matrices, zero matrices

BThe diagonal entries in A are a11, a22, a33, . . .

A diagonal matrix is a square matrix whose non-diagonal entriesare zeroes.

Matrices [1 00 2

],

1 0 00 0 00 0 2

,π 0 0

0√

2 00 0 3

are all diagonal.

Zero matrix 0 is a m × n matrix whose entries are all zero.



Sums

CIf

A =[a1 a2 · · · an

]and B =

[b1 b2 · · · bn

]are m × n matrices then

A + B =[a1 + b1 a2 + b2 · · · an + bn

]

=

a11 + b11 · · · a1j + b1j · · · a1n + b1n

......

...ai1 + bi1 · · · aij + bij · · · ain + bin

......

...am1 + bm1 · · · amj + bmj · · · amn + bmn



Scalar multiple

If c is a a scalar then

cA =[ca1 ca2 · · · can

]

=

ca11 · · · ca1j · · · ca1n

......

...cai1 · · · caij · · · cain

......

...cam1 · · · camj · · · camn



Theorem: properties of matrix addition

DLet A, B, and C be matrices of the same size and r and s bescalars.

1. A + B = B + A

2. (A + B) + C = A + (B + C )

3. A + 0 = A

4. r(A + B) = rA + rB

5. (r + s)A = rA + sA

6. r(sA) = (rs)A.


Lecture 10: Matrix multiplication. Inverse matrices

Composition of linear transformationsA

Let B be an m × n matrix, A an p ×m matrix.They define linear transformations

T : Rn −→ Rm, x 7→ Bx

andS : Rm −→ Rp, y 7→ Ay.

Their composition

(S ◦ T )(x) = S(T (x))

is a linear transformation

S ◦ T : Rn −→ Rp.

What is its matrix?



Multiplication of matricesB

We need to compute A(Bx) in matrix form. Observe

Bx = x1b1 + · · ·+ xnbn.

Hence

A(Bx) = A(x1b1 + · · ·+ xnbn)

= A(x1b1) + · · ·+ A(xnbn)

= x1A(b1) + · · ·+ xnA(bn)

=[Ab1 Ab2 · · · Abn

]x

Therefore multiplication by the matrix[Ab1 Ab2 · · · Abn

]transforms x into A(Bx).



Definition: Matrix multiplication

CIf A is an p×m matrix and B is an m× n matrix with columns b1,. . . , bn then the product AB is the p × n matrix whose columnsare Ab1, . . . , Abn:

AB = A[b1 · · · bp

]=[Ab1 · · · Abp

].



Columns of AB

DEach column Abj of AB is a linear combination of columns of Awith weights taken from the jth column of B:

Abj =[a1 · · · an

] b1j...bnj

= b1ja1 + · · ·+ bnjan



Mnemonic rules

E

[m × n matrix] · [n × p matrix] = [m × p matrix]

F

columnj(AB) = A · columnj(B)

rowi (AB) = rowi (A) · B



Theorem: Properties of matrix multiplication

GLet A be an m × n matrix and let B and C be matrices for whichindicated sums and products are defined.

1. A(BC ) = (AB)C

2. A(B + C ) = AB + AC

3. (B + C )A = BA + CA

4. r(AB) = (rA)B = A(rB) for any scalar r

5. ImA = A = AIn



Powers of matrix

H

Ak = A · · ·A (k times)

If A 6= 0 then we setA0 = I



The transpose of a matrix

IThe transpose AT of an m × n matrix A is the n ×m matrixwhose rows are formed from corresponding columns of A:

[1 2 34 5 6

]T=

1 42 53 6



Theorem: Properties of transpose

J Let A and B denote matrices matching sizes.

1. (AT )T = A

2. (A + B)T = AT + BT

3. (rA)T = r(AT ) for any scalar r

4. (AB)T = BTAT


Lectures 11–12: Characterizations of invertible matrices

Invertible matrices

A An n × n matrix A is invertible if there is an n × n matrix Csuch that

CA = I and AC = I

C is called the inverse of A.

The inverse of A, if exists, is unique (!) and is denoted A−1:

A−1A = I and AA−1 = I .



Singular matrices

BA non-invertible matrix is called a singular matrix.

An invertible matrix is nonsingular.



Theorem: Inverse of a 2× 2 matrix

CLet

A =

[a bc d

]If ad − bc 6= 0 then A is invertible and[

a bc d

]−1

=1

ad − bc

[d −b−c a

]The quantity ad − bc is called the determinant of A:

detA = ad − bc



Theorem: Solving matrix equations

DIf A is an invertible n × n matrix, then for each b ∈ Rn, theequation Ax = b has the unique solution

x = A−1b.



Quiz

E

I Suppose the second column of B is all zeroes. What can yousay about the second column of AB?



Theorem: Properties of invertible matrices

F

(a) If A is an invertible matrix, then A−1 is also invertible and

(A−1)−1 = A

(b) If A and B are n × n invertible matrices, then so is AB, and

(AB)−1 = B−1A−1

(c) If A is an invertible matrix, then so is AT , and

(AT )−1 = (A−1)T




G


(A−1)−1 = A


(AB)−1 = B−1A−1


(AT )−1 = (A−1)T




H


(A−1)−1 = A


(AB)−1 = B−1A−1


(AT )−1 = (A−1)T



Elementary matrices

IAn elementary matrix is obtained by performing a singleelementary row operation on an identity matrix.

J Theorem. If an elementary row transformation is performedon an n ×m matrix A, the resulting matrix can be written as EA,where E is made by the same row operations on In.

K Theorem. Each elementary matrix E is invertible. Theinverse of E is the elementary matrix of the same type thattransforms E back into I .



Theorem: Characterisation of invertible matrices

L An n × n matrix A is invertible

iff

A is row equivalent to In, and in this case, any sequence ofelementary row operations that reduces A to In also transforms Ininto A−1.



Computation of inverses

M

I Form the augmented matrix[A I

]and row reduce it.

I If A is row equivalent to I , then[A I

]is row equivalent to[

I A−1].

I Otherwise A has no inverse.



The Invertible Matrix Theorem 2.3.8:

NFor an n × n matrix A, the following are equivalent:

(a) A is invertible.

(b) A is row equivalent to In.

(c) A has n pivot positions.

(d) Ax = 0 has only the trivial solution.

(e) The columns of A are linearly independent.



The Invertible Matrix Theorem, continued

O

(f) x 7→ Ax is one-to-one.

(g) Ax = b has at least one solution for each b ∈ Rn.

(h) The columns of A span Rn.

(i) x 7→ Ax maps Rn onto Rn.

(j) There is an n × n matrix C such that CA = I .

(k) There is an n × n matrix D such that AD = I .

(l) AT is invertible.



One-sided inverse is the inverse

PLet A and B be square matrices.

If AB = I then both A and B are invertible and

B = A−1 and A = B−1.



Theorem: Invertible linear transformations

QLet T : Rn −→ Rn be a linear transformation and A its standardmatrix.

Then T is invertible iff A is an invertible matrix.

In that case, the linear transformation S(x) = A−1x is the onlytransformation satisfying

S(T (x)) = x for all x ∈ Rn

T (S(x)) = x for all x ∈ Rn

10212 linear algebra b - university of manchester · 2020-01-29 · 10212 linear algebra b textbook...

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