Do you feel lost when listening to native speakers? It’s likely because textbook English isn’t enough. Welcome to Native Expressions Ep.3. Real conversations are full of idioms, phrasal verbs, and cultural nuances. This series curates authentic expressions directly from podcasts and YouTube…
Are you finding it hard to tackle the Oxford 5000 word list on your own? Welcome to Oxford 5000 Intermediate Ep.10. These are the essential words you need for fluent communication. This series breaks them down step-by-step, categorized by difficulty level, to…
Do you feel lost when listening to native speakers? It’s likely because textbook English isn’t enough. Welcome to Native Expressions Ep.2. Real conversations are full of idioms, phrasal verbs, and cultural nuances. This series curates authentic expressions directly from podcasts and YouTube…
Are you finding it hard to tackle the Oxford 5000 word list on your own? Welcome to Oxford 5000 Beginner Ep.10. These are the essential words you need for fluent communication. This series breaks them down step-by-step, categorized by difficulty…
Do you feel lost when listening to native speakers? It’s likely because textbook English isn’t enough. Welcome to Native Expressions Ep.1. Real conversations are full of idioms, phrasal verbs, and cultural nuances. This series curates authentic expressions directly from podcasts…
Are you finding it hard to tackle the Oxford 5000 word list on your own? Welcome to Oxford 5000 Intermediate Ep.9. These are the essential words you need for fluent communication. This series breaks them down step-by-step, categorized by difficulty level, to…
Are you finding it hard to tackle the Oxford 5000 word list on your own? Welcome to Oxford 5000 Beginner Ep.9. These are the essential words you need for fluent communication. This series breaks them down step-by-step, categorized by difficulty…
Linear Transformation is essentially a specific rule for moving vectors from one vector space to another vector space while keeping the underlying structure intact. In this post, we will break down the formal definition, look at real-world examples (including Linear…
If you are diving into linear algebra, you have likely noticed a shift. We start with relatively comfortable grids of numbers (matrix), but suddenly, we find ourselves in the world of “abstract vector spaces” (\(V\) and \(W\)). It can feel…
Are you finding it hard to tackle the Oxford 5000 word list on your own? Welcome to Oxford 5000 Intermediate Ep.8. These are the essential words you need for fluent communication. This series breaks them down step-by-step, categorized by difficulty level, to…