Linear algebra, vector spaces, eigenvalues and eigenvectors, and linear applications.

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From 26 € /h

List of lessons

1. Vector Spaces & Subspaces
Key concepts: Understanding the definition of vector space and the ten conditions (Vector Space Axioms) to prove that a set with addition and multiplication operations in its standard form a vector space.
Linear Independence & Dependence: Testing whether vectors are linearly independent or dependent on each other.
Span, Basis & Dimension:
Determine the generative span of the space.
Finding the basis and dimension of different spaces (such as vector spaces \mathbb{R}^n, matrices M_{m \times n}, and polynomials P_n).
Subspaces: Subspace conditions, finding the null space, column space, and row space of matrices.
2. Linear Transformations and Applications
Definition and properties: Proving the linearity of applications and determining the Standard Matrix Representation.
Kernel and Range:
Calculate the kernel (Kernel/Null Space) and image (Image/Range) for conversion.
Applying the Rank-Nullity Theorem and calculating rank and nihilism.
Change of Basis: Transition Matrices and how to represent the same linear transformation under different bases and families.
3. Eigenvalues & Eigenvectors
Characteristic Equation: Finding the eigenvalues of \lambda by solving the equation \det(A - \lambda I) = 0.
Eigenvectors and Eigenspaces: Finding the eigenspace associated with each eigenvalue by solving the homogeneous system (A - \lambda I)\vec{v} = \mathbf{0}.
Diagonalization:
Diagonalizability Condition Test.
Formulating the decomposition of diagonal A = PDP^{-1} to facilitate the calculation of matrix powers A^k and the solving of systems of differential equations.
Orthogonal Diagonalization: For symmetric matrices and the application of Spectral Theorem.

Location

Online from Egypt

About Me

I am a university teaching assistant and economic researcher specializing in government research centers. I provide simplified explanations of econometrics, statistics, finance, and economics, emphasizing in-depth understanding and practical application. I help students grasp concepts, complete assignments, and prepare for exams. I also offer training in using EViews, Stata, SPSS, and Excel, as well as data analysis, interpreting statistical results, and writing academic papers. I ensure that sessions are interactive and tailored to each student's needs, with ongoing support until optimal results are achieved.

Education

I am a university lecturer specializing in econometrics, statistics, economics, finance, and business administration. I provide professional educational sessions for students of public universities, private universities, and foreign universities, as well as graduate students and researchers.

Experience / Qualifications

Bachelor of Statistics from the Faculty of Economics and Political Science, Cairo University
Experience in teaching at more than one private Egyptian university, as well as in the research field at governmental and private research centers.

Age

Teenagers (13-17 years old)

Adults (18-64 years old)

Seniors (65+ years old)

Student level

Beginner

Intermediate

Advanced

Duration

60 minutes

The class is taught in

English

Arabic

Skills

Math

Numerical analysis

Algebra

Availability of a typical week

(GMT -04:00)

New York

Online via webcam

Mon

Tue

Wed

Thu

Fri

Sat

Sun

00-04

04-08

08-12

12-16

16-20

20-24

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Summaries and worksheets to review key concepts before tests.
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Descriptive and Inferential Statistics.
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Intensive reviews before midterm and end-of-term exams.
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Welcome to my profile.
I am an experienced mathematics and science teacher with over seven years of experience helping students strengthen their understanding, improve their results and become more confident, independent learners.
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◆ MY TEACHING APPROACH ◆

• • Initial assessment • •
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• • Active problem-solving • •
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• • Targeted practice • •
Exercises are selected to address specific weaknesses, consolidate knowledge and prepare effectively for tests and examinations.

• • Regular review and feedback • •
Previously studied material is revisited when necessary, and clear feedback is provided to help the student understand what has improved and what still requires attention.

◆ WHAT YOU CAN EXPECT ◆

Professionalism and reliability are central to my work:
• • Lessons prepared in advance and tailored to the student
• • Punctual and well-structured sessions
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• • Constructive and timely feedback
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My objective is not simply to help students complete exercises. I aim to help them understand the underlying concepts, develop stronger reasoning skills and gain the confidence required to work more independently.

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Personalised Learning: Every learner is unique, and I believe in tailoring my teaching methods to suit each student's individual needs. I don't judge; instead, I concentrate on understanding their strengths and areas for improvement to create a supportive and encouraging learning environment.

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I offer one-to-one Maths tuition for GCSE, A-Level, and university students, both online and in person around Birmingham. Whether your child is struggling with the basics, preparing for upcoming exams, or aiming for top grades, I tailor every lesson to where they actually are — not where a textbook assumes they should be.

What I cover:

GCSE Maths (all exam boards — AQA, Edexcel, OCR)
A-Level Maths and Further Maths
University-level Maths for engineering and science students
Specific topics: algebra, trigonometry, calculus, statistics, mechanics, logic

How I teach:
I break difficult concepts down into simple steps, use plenty of worked examples, and give students clear practice between sessions. I'm patient with mistakes, encouraging when things feel hard, and honest about what needs work. Lessons are structured but relaxed students should feel comfortable asking "silly" questions, because those are usually the most important ones.
Exam season is right now, so if your child has GCSEs or A-Levels coming up, the next few weeks are the most important time to get focused support. I have limited slots available and can usually start within a few days.

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Many students lose confidence in math over the school year - not from lack of ability, but from pace, gaps, or one bad unit that never got resolved. This summer program (through mid-September) is built to fix that before the new year starts, not just review generally.
I work with middle and high school students, including IB, to close specific gaps not a broad refresher, but targeted work on exactly what's holding a student back. I bring the same structured, evidence-based approach I use with my university-admission students: identify precisely what's missing, build a plan around it, and track real progress, not just hours logged.
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Mathematician and higher education educator with more than four years of university teaching experience at Sapienza University of Rome, over 500 certified teaching hours and extensive undergraduate assessment experience. Additional university teaching and tutoring experience at Roma Tre University and the University of Trieste.

Experienced in face-to-face and online delivery in English and Italian, with teaching strengths in calculus, linear algebra, geometry, probability and mathematics for STEM and biomedical programmes.

Over 3 years of experience in preparing students to face AQA and Edexcel advanced level exam board (pure math, statistics and mechanics, further math)

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I am a PhD physicist and teacher known for being able to explain the most complex topics in the simplest way, from Newton's laws to the high-school student, to quantum physics to the university student.

Teaching focus:
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- Problem solving: breaking a question down into its simplest parts, visualising the physics involved.

Format:
- Lessons at my home where I have a whiteboard and desk set up for teaching
- Online via Google meet
- Lessons at the student home (depending on distance, possible surcharge)

Exams: Any pre-university examination, including Matura, PAM, IB, A-levels. The type of exam is not as important as the topics being studied.

What students say:
- "I had to repeat the first semester of medical school because of physics, and after working with Ralph, physics actually became one of my best subjects. He explained the theory incredibly clearly and went through the exercises step by step. I could ask questions at any time, and I was also able to decide what pace we wanted to work at during our lessons." (Helena, UZH student)

Personal accomplishments:
- Gold medalist at the International Astronomy Olympiad (Beijing 2005)
- Lecturer at ETH Zurich 2020-2024
- Phd in Theoretical Physics, University of Bristol
- B.Tech. in Engineering Physics, IIT Bombay