Preimage of a set | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

Transformations in mathematics involve mapping elements from one set to another, while pre-images represent the set of vectors in the domain that map to a specific subset in the codomain.

Transcript

Let's add some transformation that maps elements in set X to set Y. We know that we call X the domain of T. So that's my set X and then my set that I'm mapping into, set Y, that's the codomain. We know that T is a transformation that if you take any member of X and you transform it, you'll associate it with a member of set Y. You'll map it to a mem... Read More

Key Insights

• 😫 A transformation, or function, in mathematics maps elements from one set (domain) to another set (codomain).
• 😫 The image of a subset A under a transformation T is the set of all transformations of the members in A.
• 😮 The pre-image of a subset S under a transformation T is the set of vectors in the domain that map into S.
• 🎅 The pre-image of a subset S may not include all elements in S, but only the vectors in the domain that map into S.
• 🎅 The inverse T of S is equal to the pre-image of S under T.
• 😑 The pre-image can be viewed as a cancellation of the image when constructing a subset of S.
• 😑 The inverse notation represents the pre-image and was likely introduced to indicate the cancellation of the image.

Questions & Answers

Q: What is a transformation in mathematics?

A transformation, also known as a function, maps elements from one set (domain) to another set (codomain) in mathematics.

Q: How is the image of a subset defined under a transformation?

The image of a subset A under a transformation T is the set of all transformations of the members in A, which are subsets of the codomain.

Q: What is the pre-image of a subset in the codomain under a transformation?

The pre-image of a subset S in the codomain under a transformation T is the set of vectors in the domain that map into S.

Q: Do all elements in a subset S necessarily get mapped to from the domain?

No, the pre-image of a subset S only includes the vectors in the domain that map into S, but not necessarily all elements in S.

Summary & Key Takeaways

• A transformation, also called a function, maps elements from one set (domain) to another set (codomain).

• The image of a subset A in the domain under a transformation T is the set of all transformations of the members in A.

• The pre-image of a subset S in the codomain under a transformation T is the set of vectors in the domain that map into S.