Vector transformations | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Functions in linear algebra are mappings from one set to another, and vectors can be valid elements of sets. A transformation is a function that operates on vectors, changing one vector into another.

Transcript

In the last video we saw, a little bit more formally than you might have been exposed to the past, that a function is just a mapping of the members of one set to another set. So if this is my first set, x. We call that the domain. And the set that we're mapping to, y, in this case, that's called the codomain. And the function just maps each of the ... Read More

Key Insights

• 😫 Functions in linear algebra are mappings of sets, where specific entries of one set are associated with entries in another set.
• 😫 Vectors are valid elements of sets, and in linear algebra, they are often represented as column vectors with real number components.
• 🤙 Functions that operate on vectors, changing one vector into another, are called transformations in linear algebra.

Questions & Answers

Q: What is a function in linear algebra?

In linear algebra, a function is a mapping of the members of one set to another set, where specific entries of the first set are associated with entries in the second set.

Q: How are vectors represented in linear algebra?

Vectors in linear algebra are often represented as column vectors, an ordered list of n-components where each component is a member of the real numbers.

Q: What is a transformation in linear algebra?

A transformation is a function that operates on vectors, changing one vector into another. It is often used interchangeably with the term "function" in the context of linear algebra.

Q: How do functions and transformations relate to video game programming?

Functions and transformations are essential in video game programming as they allow for the manipulation and transformation of images or objects based on various factors such as angle and position.

Summary & Key Takeaways

• Functions in linear algebra are mappings of the members of one set to another set, often represented by a function that associates specific entries of one set with entries in another set.

• Vectors in linear algebra are members of sets, particularly in rn, which represents the set of all n-tuples where each component is a real number.

• Functions that map vectors from one set to another are referred to as transformations, and they play a significant role in linear algebra and applications like video game programming.