10212 Linear Algebra B 10212 Linear Algebra B University of Manchester 27 January 2020
10212 Linear Algebra B
10212 Linear Algebra B
University of Manchester
27 January 2020
10212 Linear Algebra B
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
What linear Algebra is about?
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
10212 Linear Algebra B
What linear Algebra is about?
Compression of notation
C
I . . . and then abbreviated
[p1 p2 p3
] g1g2g3
= PTG ,
where
P =
p1p2p3
and G =
g1g2g3
10212 Linear Algebra B
What linear Algebra is about?
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.
10212 Linear Algebra B
What linear Algebra is about?
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.
10212 Linear Algebra B
What linear Algebra is about?
Prerequisites:
More abstract bits of MATH10111:
I Functions: 1–1, onto,bijective.
I Equivalence relations.
I Binary operations and groups.
10212 Linear Algebra B
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.
10212 Linear Algebra B
Lecture 1: Systems of linear equations
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
10212 Linear Algebra B
Lecture 1: Systems of linear equations
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
I A matrix with m rows and n columns is called an m× nmatrix.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
Lecture 2: Elementary operations
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.
10212 Linear Algebra B
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?
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
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).
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
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.
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
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.
10212 Linear Algebra B
Lecture 3: Row reduction and 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).
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
Examples
G The following is a schematic presentation of an echelonmatrix: � ∗ ∗ ∗ ∗
0 � ∗ ∗ ∗0 0 0 � ∗
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
H and this is a reduced echelon matrix:1 0 ∗ 0 ∗0 1 ∗ 0 ∗0 0 0 1 ∗
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
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.
10212 Linear Algebra B
Lecture 3: Row reduction and Echelon form
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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.
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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.
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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.
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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
10212 Linear Algebra B
Lecture 4: Solution of Linear Systems
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.
10212 Linear Algebra B
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.
10212 Linear Algebra B
Lecture 5: Vector equations
Vectors
B 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.
10212 Linear Algebra B
Lecture 5: Vector equations
Vectors
C 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.
10212 Linear Algebra B
Lecture 5: Vector equations
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
.
10212 Linear Algebra B
Lecture 5: Vector equations
Operations on vectorsE
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
.
10212 Linear Algebra B
Lecture 5: Vector equations
Operations on vectorsF
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
.
10212 Linear Algebra B
Lecture 5: Vector equations
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.)
10212 Linear Algebra B
Lecture 5: Vector equations
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.
10212 Linear Algebra B
Lecture 5: Vector equations
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
10212 Linear Algebra B
Lecture 5: Vector equations
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.
10212 Linear Algebra B
Lecture 5: Vector equations
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.
10212 Linear Algebra B
Lecture 5: Vector equations
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]
10212 Linear Algebra B
Lecture 5: Vector equations
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.
10212 Linear Algebra B
Lecture 5: Vector equations
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.
10212 Linear Algebra B
Lecture 5: Vector equations
Span
O
Span{v1, . . . , vp} = {b : x1v1 + · · · xpvp = b
has a solution }
10212 Linear Algebra B
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
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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
.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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
.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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].
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = 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.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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).
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
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.
10212 Linear Algebra B
Lecture 6: The matrix equation Ax = b
Example
M
x1 + 2x2 − x3 = 0
x1 + 3x3 + x3 = 0
10212 Linear Algebra B
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
10212 Linear Algebra B
Lecture 7: Linear independence
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.
10212 Linear Algebra B
Lecture 7: Linear independence
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.
10212 Linear Algebra B
Lecture 7: Linear independence
Linear dependence
EThe set
{ v1, . . . , vp }
in Rn is linearly dependent if there exist weights c1, . . . , cp, notall zero, such that
c1v1 + · · ·+ cpvp = 0
10212 Linear Algebra B
Lecture 7: Linear independence
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
10212 Linear Algebra B
Lecture 7: Linear independence
Linear independence of matrix columns
GThe columns of matrix A are linearly independent iff the equation
Ax = 0
has only the trivial solution.
10212 Linear Algebra B
Lecture 7: Linear independence
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.
10212 Linear Algebra B
Lecture 7: Linear independence
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.
10212 Linear Algebra B
Lecture 7: Linear independence
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.
10212 Linear Algebra B
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 .
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
Matrix transformations
B
T : Rn −→ Rm
x 7→ Ax
where A is an m × n matrix.
In short:T (x) = Ax.
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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 .
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
Properties of linear transformations
EIf T is a linear transformation then
T (0) = 0
andT (cu + dv) = cT (u) + dT (v).
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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 .
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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 .
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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.
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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.
10212 Linear Algebra B
Lecture 8: Introduction to Linear Transformations
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.
10212 Linear Algebra B
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
10212 Linear Algebra B
Lecture 9: Matrix operations
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.
10212 Linear Algebra B
Lecture 9: Matrix operations
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
10212 Linear Algebra B
Lecture 9: Matrix operations
Scalar multiple
If c is a a scalar then
cA =[ca1 ca2 · · · can
]
=
ca11 · · · ca1j · · · ca1n
......
...cai1 · · · caij · · · cain
......
...cam1 · · · camj · · · camn
10212 Linear Algebra B
Lecture 9: Matrix operations
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.
10212 Linear Algebra B
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?
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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).
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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
].
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
Mnemonic rules
E
[m × n matrix] · [n × p matrix] = [m × p matrix]
F
columnj(AB) = A · columnj(B)
rowi (AB) = rowi (A) · B
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
Powers of matrix
H
Ak = A · · ·A (k times)
If A 6= 0 then we setA0 = I
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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
10212 Linear Algebra B
Lecture 10: Matrix multiplication. Inverse matrices
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
10212 Linear Algebra B
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 .
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
Singular matrices
BA non-invertible matrix is called a singular matrix.
An invertible matrix is nonsingular.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
Quiz
E
I Suppose the second column of B is all zeroes. What can yousay about the second column of AB?
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
Theorem: Properties of invertible matrices
G
(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
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
Theorem: Properties of invertible matrices
H
(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
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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 .
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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.
10212 Linear Algebra B
Lectures 11–12: Characterizations of invertible matrices
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