im(T): Image of a transformation | Matrix transformations | Linear Algebra | Khan Academy

TL;DR

This video explains the concept of subspaces in linear algebra and explores the image of a subspace under a transformation.

Transcript

Let's say that I have some set V that is a subspace in Rn. And just as a reminder, what does it mean? That's just some set, or some subset of Rn where if I take any two members of that subset-- so let say I take the members a and b-- they're both members my subspace. By the fact that this is a subspace, we then know that the addition of these two v... Read More

Key Insights

• 😫 A subspace is a set in Rn that satisfies certain properties, such as closure under vector addition and scalar multiplication.
• 😫 The image of a subspace under a transformation is the set of vectors obtained by applying the transformation to each member of the subspace.
• ❓ The image of Rn under a transformation represents all possible values that the transformation can output.

Questions & Answers

Q: What is a subspace and what are the properties it must satisfy?

A subspace is a set in Rn where the addition of any two vectors and the multiplication of any vector by a scalar always result in vectors that remain in the subspace. It must also contain the zero vector.

Q: How does a transformation affect a subspace?

When a transformation is applied to a subspace, the resulting image is a set of vectors obtained by transforming each member of the subspace. The image may or may not be a subspace, depending on the properties of the transformation.

Q: What is the image of Rn under a transformation?

The image of Rn under a transformation represents all the vectors in the codomain that are mapped to by the transformation. It is the set of all possible values that the transformation can output when given any vector from Rn.

Q: What is the relationship between the image of a transformation and the column space of its matrix representation?

The image of a transformation is equivalent to the column space of its matrix representation. The column space represents all the linear combinations of the column vectors in the matrix, which is essentially the set of all possible outputs of the transformation.

Summary & Key Takeaways

• A subspace is a set in Rn where any two members in the set, when added together or multiplied by a scalar, remain in the subspace.

• The image of a subspace under a transformation is the set of all vectors obtained by transforming each member of the subspace.

• The image of Rn under a transformation represents all the vectors in the codomain that are mapped to by the transformation.