2DBN00 - Linear Algebra
A Summary
Fe Fan Li
October 20, 2020
, Contents
Preface 1
1 Matrices and Systems of Equations 3
1.1 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Row echelon form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Matrx algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Final exam - 2019-07 - Question 1a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Determinants 8
2.1 The determinant of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Properties of determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Final Exam - 2018-04 - Question 1b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Vector Spaces 10
3.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.6 Row space and column space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.7 Final Exam - 2018-07 - Quetion 1b,c and d . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Linear Transformation 16
5 Orthogonality 17
5.1 The scalar product in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Orthogonal subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Least squares problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.4 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.6 The Gram-Schmidt orthogonalization process . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.7 Final Exam - 2019-07 - Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Eigenvalues 24
6.1 Systems of linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.3 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.4 Final Exam - 2019-07 - Question 1b and 1c . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
,Preface
This summary is based on the course 2DBN00, its lectures and the book Linear Algebra with Applications
by Steven J. Leon. Use this summary at your own risk; I am not responsible for your exam
and sections in this summary may contain errors, redundant information or absence of
important information. Information may be outdated and not part of the course anymore.
My recommendation is that you also study the book, lecture slides and exercises as well
as exam questions thoroughly.
The structure of this document follows the order of the chapters in the book in the order that they are
treated during lectures. The mathematics of complex numbers will not be treated. There will be a small
example exam question after every chapter, but that is by no means sufficient. It is useful to check
out all the examples with elaborations in the book for this course, but it would defeat the purpose of a
summary too much if I post them here, I will only outline the definitions and theorems and other pieces
of information along with small exam examples.
2
,Chapter 1
Matrices and Systems of Equations
1.1 Systems of linear equations
Equivalent systems
Definition: Two systems of equations involving the same variables are said to be equivalent if they
have the same solution set.
There are three operations that can be used on a system to obtain an equivalent system:
1. The order in which any two equations are written may be interchanged.
2. Both sides of an equation may be multiplied by the same nonzero real number.
3. A multiple of one equation may be added to (or subtracted from) another.
n × n systems
Definition: A system is said to be in strict triangular form if, in the kth equation, the coefficients
of the first k − 1 variables are all zero and the coefficient xk is nonzero (k = 1, . . . , n).
Elementary row operations:
1. Interchange two rows.
2. Multiply a row by a nonzero real number.
3. Replace a row by its sum with a multiple of another row.
1.2 Row echelon form
Definition: A matrix is said to be in row echelon form if
1. The first nonzero entry in each nonzero row is 1.
2. If row k does not consist entirely of zeros, the number of leading zero entries in row k + 1 is greater
than the number of leading zero entries in row k.
3. If there are rows whose entries are all zero, they are below the rows having nonzero entries.
Definition: The process of using row operations 1,2 and 3 to transform a linear system into one whose
augmented matrix is in row echelon form is called Gaussian elimination.
Overdetermined systems
A linear system is said to be overdetermined if there are more equations than unknowns. Overdetermined
systems are usually (but not always) inconsistent.
3
, Underdetermined systems
A system of m linear equations and n unknowns is said to be underdetermined if there are fewer equations
than unknowns (m < n). Although it is possible for underdetermined systems to be inconsistent, they
are usually consistent with infinitely many solutions. It is not possible for an underdetermined system
to have a unique solution.
Reduced row echelon form
Definition: A matrix is said to be in reduced row echelon form if
1. The matrix is in row echelon form.
2. The first nonzero entry in each row is the only nonzero entry in its column.
The process of using elementary row operations to transform a matrix into reduced row echelon form is
called Gauss-Jordan reduction.
Homogeneous systems
A system of linear equations is said to be homogeneous if the constants on the right hand side are all
zero. Homogeneous systems are always consistent.
Theorem 1.2.1 An m × n homogeneous system of linear equations has a nontrivial solution if n > m.
1.3 Matrix arithmetic
Read the book on how matrix and vector notation is defined.
Equality
Definition: Two m × n matrices A and B are said to be equal if aij = bij for each i and j.
Scalar multiplication
Definition: If A is an m × n matrix and α is a scalar, then αA is the m × n matrix whose (i, j) entry is
αaij .
Matrix addition
Definition: If A = (aij and B = (bij are both m × n matrices, then the sum A + B is the m × n matrix
whose (i, j) entry is aij + bij for each ordered pair (i, j).
Matrix multiplication and linear systems
Definition: If a1 , . . . , an are vectors in Rm and c1 , c2 , . . . , cn are scalars, then a sum of the form
c1 a1 + · · · + cn an
is said to be a linear combination of the vectors a1 , . . . , an .
If A is an m × n matrix and x is a vector in Rn , then
Ax = x1 a1 + · · · + xn an
Theorem 1.3.1 Consistency theorem for linear systems
A linear system Ax = b is consistent if and only if b can be written as a linear combination of the column
vectors of A.
Matrix multiplication
Definition: If A = (aij ) is an m × n matrix and B = (bij ) is an n × r matrix, then the product
AB = C = (cij ) is the m × r matrix whose entries are defined by
n
X
cij = ~
ai bj = aik bkj
k=1
4
A Summary
Fe Fan Li
October 20, 2020
, Contents
Preface 1
1 Matrices and Systems of Equations 3
1.1 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Row echelon form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Matrx algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Final exam - 2019-07 - Question 1a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Determinants 8
2.1 The determinant of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Properties of determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Final Exam - 2018-04 - Question 1b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Vector Spaces 10
3.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Linear independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Basis and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.6 Row space and column space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.7 Final Exam - 2018-07 - Quetion 1b,c and d . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Linear Transformation 16
5 Orthogonality 17
5.1 The scalar product in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Orthogonal subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Least squares problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.4 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.6 The Gram-Schmidt orthogonalization process . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.7 Final Exam - 2019-07 - Question 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Eigenvalues 24
6.1 Systems of linear differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.3 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.4 Final Exam - 2019-07 - Question 1b and 1c . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1
,Preface
This summary is based on the course 2DBN00, its lectures and the book Linear Algebra with Applications
by Steven J. Leon. Use this summary at your own risk; I am not responsible for your exam
and sections in this summary may contain errors, redundant information or absence of
important information. Information may be outdated and not part of the course anymore.
My recommendation is that you also study the book, lecture slides and exercises as well
as exam questions thoroughly.
The structure of this document follows the order of the chapters in the book in the order that they are
treated during lectures. The mathematics of complex numbers will not be treated. There will be a small
example exam question after every chapter, but that is by no means sufficient. It is useful to check
out all the examples with elaborations in the book for this course, but it would defeat the purpose of a
summary too much if I post them here, I will only outline the definitions and theorems and other pieces
of information along with small exam examples.
2
,Chapter 1
Matrices and Systems of Equations
1.1 Systems of linear equations
Equivalent systems
Definition: Two systems of equations involving the same variables are said to be equivalent if they
have the same solution set.
There are three operations that can be used on a system to obtain an equivalent system:
1. The order in which any two equations are written may be interchanged.
2. Both sides of an equation may be multiplied by the same nonzero real number.
3. A multiple of one equation may be added to (or subtracted from) another.
n × n systems
Definition: A system is said to be in strict triangular form if, in the kth equation, the coefficients
of the first k − 1 variables are all zero and the coefficient xk is nonzero (k = 1, . . . , n).
Elementary row operations:
1. Interchange two rows.
2. Multiply a row by a nonzero real number.
3. Replace a row by its sum with a multiple of another row.
1.2 Row echelon form
Definition: A matrix is said to be in row echelon form if
1. The first nonzero entry in each nonzero row is 1.
2. If row k does not consist entirely of zeros, the number of leading zero entries in row k + 1 is greater
than the number of leading zero entries in row k.
3. If there are rows whose entries are all zero, they are below the rows having nonzero entries.
Definition: The process of using row operations 1,2 and 3 to transform a linear system into one whose
augmented matrix is in row echelon form is called Gaussian elimination.
Overdetermined systems
A linear system is said to be overdetermined if there are more equations than unknowns. Overdetermined
systems are usually (but not always) inconsistent.
3
, Underdetermined systems
A system of m linear equations and n unknowns is said to be underdetermined if there are fewer equations
than unknowns (m < n). Although it is possible for underdetermined systems to be inconsistent, they
are usually consistent with infinitely many solutions. It is not possible for an underdetermined system
to have a unique solution.
Reduced row echelon form
Definition: A matrix is said to be in reduced row echelon form if
1. The matrix is in row echelon form.
2. The first nonzero entry in each row is the only nonzero entry in its column.
The process of using elementary row operations to transform a matrix into reduced row echelon form is
called Gauss-Jordan reduction.
Homogeneous systems
A system of linear equations is said to be homogeneous if the constants on the right hand side are all
zero. Homogeneous systems are always consistent.
Theorem 1.2.1 An m × n homogeneous system of linear equations has a nontrivial solution if n > m.
1.3 Matrix arithmetic
Read the book on how matrix and vector notation is defined.
Equality
Definition: Two m × n matrices A and B are said to be equal if aij = bij for each i and j.
Scalar multiplication
Definition: If A is an m × n matrix and α is a scalar, then αA is the m × n matrix whose (i, j) entry is
αaij .
Matrix addition
Definition: If A = (aij and B = (bij are both m × n matrices, then the sum A + B is the m × n matrix
whose (i, j) entry is aij + bij for each ordered pair (i, j).
Matrix multiplication and linear systems
Definition: If a1 , . . . , an are vectors in Rm and c1 , c2 , . . . , cn are scalars, then a sum of the form
c1 a1 + · · · + cn an
is said to be a linear combination of the vectors a1 , . . . , an .
If A is an m × n matrix and x is a vector in Rn , then
Ax = x1 a1 + · · · + xn an
Theorem 1.3.1 Consistency theorem for linear systems
A linear system Ax = b is consistent if and only if b can be written as a linear combination of the column
vectors of A.
Matrix multiplication
Definition: If A = (aij ) is an m × n matrix and B = (bij ) is an n × r matrix, then the product
AB = C = (cij ) is the m × r matrix whose entries are defined by
n
X
cij = ~
ai bj = aik bkj
k=1
4
