Affine subspace

Definition

An affine subspace is a set of points that can be described as a translation of a linear subspace. It can be thought of as a geometric object that does not have an origin and can be represented in terms of a linear combination of its points, where the coefficients sum up to one. Affine subspaces play a key role in understanding the properties and behaviors of affine transformations, which include translations, rotations, and scaling in vector spaces.

5 Must Know Facts For Your Next Test

1. Affine subspaces can be defined by a point and a direction, meaning they extend infinitely in the direction of the vectors that define them.
2. Every affine subspace can be represented as an affine transformation applied to a linear subspace.
3. The dimension of an affine subspace is equal to the dimension of the corresponding linear subspace it is derived from.
4. Affine subspaces are not necessarily vector spaces because they do not include the zero vector unless the subspace itself is the origin.
5. To check if a set of points forms an affine subspace, one can verify if any two points can generate the entire set through affine combinations.

Review Questions

• How do affine subspaces differ from linear subspaces in terms of their properties?
  • Affine subspaces differ from linear subspaces primarily in that they do not have a designated origin or contain the zero vector. While a linear subspace is closed under addition and scalar multiplication and includes the zero vector, an affine subspace is formed by translating a linear subspace, thus preserving its structure but shifting it away from the origin. This distinction allows affine subspaces to represent geometric figures such as lines or planes without fixed reference points.
• Discuss how an affine transformation relates to the concept of an affine subspace and provide an example.
  • An affine transformation relates closely to an affine subspace as it modifies the positions of points in space while preserving their relationships. For example, if we take a line (an affine subspace) defined by two points A and B, applying an affine transformation such as translation will shift both points A and B to new locations without altering their relative position. This shows how the essence of an affine subspace remains intact even when transformed.
• Evaluate the significance of convex combinations in understanding the structure of affine subspaces.
  • Convex combinations are significant in understanding the structure of affine subspaces because they provide a way to express points within these spaces based on other known points. By requiring that the coefficients are non-negative and sum to one, convex combinations ensure that we remain within the bounds defined by an affine set. This concept helps in characterizing how any point in an affine subspace can be generated from other points, facilitating deeper insights into geometric interpretations and applications in various fields such as optimization and computer graphics.

Related terms

Linear Subspace: A linear subspace is a subset of a vector space that is closed under vector addition and scalar multiplication, containing the zero vector.

Affine Transformation: An affine transformation is a function between affine spaces that preserves points, straight lines, and planes. It includes linear transformations followed by translations.

Convex Combination: A convex combination is a linear combination of points where all coefficients are non-negative and sum to one, often used in defining points in an affine subspace.