MATH1048 Linear Algebra I Lecture Notes

MATH1048 Linear Algebra I Lecture Notes

Mathematical

Sciences

University of Southampton

MATH1048 Linear Algebra I

Autumn 2025-2026
Ian Leary and Ashot Minasyan

Contents

1. Notation

2. Chapter 0. Introduction to complex numbers

   1. Definition and operations

   2. Graphical representation of complex numbers

   3. Complex conjugation and reciprocals

   4. Solving polynomial equations over complex numbers

3. Chapter 1. The real n-space

   1. The real n-space

   2. Scalar Product

   3. Norm of a vector

   4. Projections and Cauchy-Schwarz inequality

   5. Equation of a line

   6. Parametric equation of a plane

   7. Cartesian equation of a plane in R3

   8. Vector product

   9. Intersections of planes and lines in R3

   10. Distances in R3

4. Chapter 2. Matrix Algebra

   1. Basic definitions and terminology

   2. Operations with Matrices

   3. The transpose of a matrix

   4. The inverse of a matrix

   5. Powers of a matrix

5. Chapter 3. Systems of Linear Equations

   1. Systems of Equations and Matrices

   2. Row operations and Gaussian elimination

   3. Matrix inverse using row operations

   4. Rank of a matrix

6. Chapter 4. Determinants

   1. Axiomatic definition of determinant

   2. Determinants and invertibility

   3. Calculating determinants using cofactors

7. Chapter 5. Linear transformations

   1. Basic definitions and properties

   2. Linear transformations of R2

   3. Composition of linear transformations

   4. The inverse of a linear transformation

8. Chapter 6. Subspaces of Rn

   1. Definition and basic examples

   2. Null spaces

   3. Linear span

9. Range and column space

10. Linear independence

    1. Bases

11. Chapter 7. Eigenvalues, eigenvectors and applications

    1. Eigenvalues and eigenvectors

    2. More examples

    3. Application of eigenvectors in Google’s page ranking algorithm

12. Symmetric matrices

    1. Orthogonal diagonalization of symmetric matrices

13. Quadratic forms

14. Chapter 8. Orthonormal sets and quadratic forms

    1. Basic notions

Notation

• $ eq$: not equal

• $ orall$: for all

• $ herefore$: therefore

• $ orall n ext{ in } N$: for every natural number n

• $ ext{dim}(V)$: the dimension of subspace V

Chapter 0. Introduction to Complex Numbers

Historically, complex numbers emerged to solve polynomial equations lacking real solutions, such as $x^2 + 1 = 0$ or $3x^8 + 10x^4 + 2 = 0$. To encompass these, we need to introduce the imaginary unit $i$, defined by $i^2 = -1$. Consequently, the set of complex numbers $C$ is formed as $z = a + bi$, where both $a$ and $b$ are real numbers.

0.1 Definition and Operations

Definition 0.1 (Complex numbers): A complex number is defined as $z = a + bi$, where:

• $a$ is the real part of $z$, denoted $Re(z)$,

• $b$ is the imaginary part of $z$, denoted $Im(z)$,

• $i$ is the imaginary unit.

The operations on complex numbers include addition and subtraction defined as follows:

• If $z = a + bi$ and $w = c + di$, then:

  • Addition: $z + w = (a + c) + (b + d)i$

  • Subtraction: $z - w = (a - c) + (b - d)i$

For instance, $(1 + 5i) + (-3 - i) = (-2 + 4i).$

0.2 Graphical Representation of Complex Numbers

Complex numbers correspond to points in the Cartesian plane, where the x-axis represents real numbers and the y-axis represents imaginary numbers. Each complex number can be represented as a vector $(a, b)$ in $ ext{R}^2$.

Definition 0.5 (Modulus of a Complex Number): The modulus of $z = a + bi$ is given by
$|z| = ext{√(a^2 + b^2)}.$

This represents the length of the vector corresponding to $z$.

Chapter 1. The Real n-Space

1.1 The Real n-Space

Definition 1.1 (The Real n-Space): The real n-space $ ext{R}^n$ consists of all n-tuples $(x1, x2, ext{…}, x_n)$.

1.2 Scalar Product

Definition 1.6 (Scalar Product): Given two vectors $u, v ext{ in } R^n$, the scalar product of $u$ and $v$ is defined by $u \bullet v = u<em>1v</em>1 + u<em>2v</em>2 + … + u<em>nv</em>n.$
It translates geometrically to the cosine of the angle between the two vectors when normalized.

1.3 Norm of a Vector

Definition 1.11 (Norm): The norm of a vector $v = (v1, v2, ext{…}, vn)$ is $ext{||v||} = ext{√(v</em>1^2 + v<em>2^2 + … + v</em>n^2)}.$

1.4 Projections and Cauchy-Schwarz Inequality

The projection of vector $u$ onto vector $v$ is given by the formula:
$ext{proj}_v u = rac{(u \bullet v)}{(v \bullet v)} v.$ The Cauchy-Schwarz inequality states that
$|u \bullet v| ext{ ≤ } ext{||u|| ||v||}.$

1.5 Equation of a Line

The equation of a line can be represented parametrically as $extbf{L} = P + t extbf{d},$ where $P$ is a point on the line, $ extbf{d}$ is the direction vector, and $t$ is a scalar.

1.6 Parametric Equation of a Plane

For plane equations, parametric representations can be made using two non-parallel vectors.

1.8 Vector Product

For vectors $a = (a1, …, an)$ and $b = (b1, …, bn)$, the vector product results in a new vector orthogonal to both $a$ and $b$, calculated as:
$a imes b = \begin{pmatrix} a<em>2 b</em>3 - a<em>3 b</em>2 \ a<em>3 b</em>1 - a<em>1 b</em>3 \ a<em>1 b</em>2 - a<em>2 b</em>1 \end{pmatrix}.$

1.9 Distances in R3

The distance between two points $A, B$ in $ ext{R}^3$ is given by $d(A, B) = ||B - A|| = ext{√((b<em>1 - a</em>1)^2 + (b<em>2 - a</em>2)^2 + (b<em>3 - a</em>3)^2)}.$

Chapter 2. Matrix Algebra

2.1 Basic Definitions and Terminology

Definition 2.1 (Matrix): An m × n matrix $A$ is an array of real or complex numbers, which allows operations analogous to number systems.

2.2 Operations with Matrices

Definition 2.4 (Basic Operations with Matrices): Let $A$ and $B$ be two matrices, the operations defined include scalar multiplication, addition, and subtraction. For example:

• For matrices $A = (a{ij})$ and $B = (b{ij})$, $A + B = (a<em>{ij} + b</em>{ij}).$

Chapter 3. Systems of Linear Equations

3.1 Systems of Equations and Matrices

Systems can be expressed as matrix equations, where coefficients are arranged in a matrix format (A|b) as mentioned previously, allowing simpler solutions.

3.2 Row Operations and Gaussian Elimination

Row Operations change the system’s augmented matrix into forms suitable for solutions. Gaussian elimination systematically applies these to arrive at reduced row echelon forms.

3.3 Matrix Inverse using Row Operations

A matrix is invertible if its augmented form can be manipulated to form $I_n$. If successful, that inverse can be derived from the operations performed.

3.4 Rank of a Matrix

The rank can be determined by the number of non-zero rows after row-reducing the matrix. Non-zero rows correspond to the vectors in the linear combinations spanning the matrix.

Chapter 4. Determinants

4.1 Axiomatic Definition of Determinant

Definition 4.1 determines conditions required for determinant function properties.

4.2 Determinants and Invertibility

Determinants denote invertibility: a non-zero determinant signals a non-singular matrix.

4.3 Calculating Determinants Using Cofactors

The determinants can be calculated through minors and cofactors; transforming matrices affects the determinant value accordingly.

Chapter 0. Introduction to Complex Numbers

Historically, complex numbers emerged to solve polynomial equations lacking real solutions, such as $x^2 + 1 = 0$. To encompass these, we introduce the imaginary unit $i$, defined by $i^2 = -1$. The set of complex numbers $\mathbb{C}$ is formed as $z = a + bi$, where $a, b \in \mathbb{R}$.

0.1 Definition and Operations

Definition 0.1 (Complex numbers): A complex number is $z = a + bi$, where $a = Re(z)$ and $b = Im(z)$.

How to perform operations:

1. Addition/Subtraction: Group the real parts and the imaginary parts separately.

   • $(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$

2. Multiplication: Use the FOIL method (distributive law) and substitute $i^2 = -1$.

   • $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

0.2 Graphical Representation and Modulus

Each complex number is represented as a vector $(a, b)$ in the complex plane.

Definition 0.5 (Modulus): The modulus represents the distance from the origin.

• Calculation: For $z = a + bi$, calculate $\text{||}z\text{||} = \sqrt{a^2 + b^2}$.

Chapter 1. The Real n-Space

1.1 Scalar Product

Definition 1.6 (Scalar Product): For $u, v \in \mathbb{R}^n$.

• Calculation: Multiply corresponding components and find their sum.

• $u \cdot v = u<em>1v</em>1 + u<em>2v</em>2 + \dots + u<em>nv</em>n$

1.2 Norm of a Vector

Definition 1.11 (Norm): The magnitude of vector $v$.

• Calculation: $\text{||}v\text{||} = \sqrt{v<em>1^2 + v</em>2^2 + \dots + v_n^2}$.

1.3 Projections and Cauchy-Schwarz Inequality

How to project $u$ onto $v$:

1. Calculate the scalar product $u \cdot v$.

2. Calculate the norm squared of $v$, which is $v \cdot v$.

3. Use the formula: $\text{proj}_v u = \frac{u \cdot v}{v \cdot v} v$.

1.4 Vector Product (Cross Product in $\mathbb{R}^3$)

How to calculate $a \times b$:
Use the determinant of a $3 \times 3$ matrix with unit vectors $i, j, k$ or the component formula:

• $a \times b = \begin{pmatrix} a<em>2 b</em>3 - a<em>3 b</em>2 \ a<em>3 b</em>1 - a<em>1 b</em>3 \ a<em>1 b</em>2 - a<em>2 b</em>1 \end{pmatrix}$

Chapter 2. Matrix Algebra

2.2 Operations with Matrices

• Addition: Add elements in the same position: $(A + B)<em>{ij} = a</em>{ij} + b_{ij}$.

• Scalar Multiplication: Multiply every entry in the matrix by the scalar $k$.

• Multiplication ($AB$): The entry in row $i$ and column $j$ is the scalar product of row $i$ of $A$ and column $j$ of $B$.

Chapter 3. Systems of Linear Equations

3.2 Row Operations and Gaussian Elimination

How to solve using Gaussian Elimination:

1. Write the system as an augmented matrix $(A|b)$.

2. Use Elementary Row Operations (EROs):

   • Swap two rows.

   • Multiply a row by a non-zero scalar.

   • Add a multiple of one row to another.

3. Aim for Row Echelon Form (REF) (zeros below pivots) or Reduced Row Echelon Form (RREF) (zeros above and below pivots, pivots are 1).

3.3 Matrix Inverse via Row Operations

How to find $A^{-1}$:

1. Form the augmented matrix $(A|I)$, where $I$ is the identity matrix.

2. Apply EROs to transform the left side into $I$.

3. The resulting right side is $A^{-1}$. If you cannot reach $I$, the matrix is not invertible.

Chapter 4. Determinants

4.3 Calculating Determinants Using Cofactors

How to calculate a determinant:

1. For a $2 \times 2$ matrix: $\det \begin{pmatrix} a &amp; b \ c &amp; d \end{pmatrix} = ad - bc$.

2. For larger matrices, use Cofactor Expansion:

   • Choose a row or column (preferably with zeros).

   • $\det(A) = \sum<em>{j=1}^n a</em>{ij} C<em>{ij}$, where $C</em>{ij} = (-1)^{i+j} \det(M<em>{ij})$ and $M</em>{ij}$ is the minor matrix.