Subspaces and the basis for a subspace

When studying linear subspaces, focus on the following key points:

1. Definition: A linear subspace is a subset of a vector space that is closed under vector addition and scalar multiplication.

2. Criteria for Subspaces:

   • Contains the zero vector.
   • Closed under addition: If $u, v$ are in the subspace, then $u + v$ is also in the subspace.
   • Closed under scalar multiplication: If $u$ is in the subspace and $c$ is a scalar, then $c \cdot u$ is also in the subspace.

3. Examples: Common examples include:

   • The zero subspace.
   • The whole vector space.
   • Lines and planes through the origin in $\mathbb{R}^n$.

4. Dimension: The dimension of a subspace is the number of vectors in a basis for that subspace, representing the number of degrees of freedom.

5. Basis and Span:

   • A basis is a set of vectors that are linearly independent and span the subspace.
   • The span of a set of vectors is the set of all linear combinations of those vectors.

6. Intersection and Sum of Subspaces:

   • The intersection of subspaces is also a subspace.
   • The sum of two subspaces consists of all sums of vectors from each subspace.

7. Linear Independence: Understanding the concept of linear independence is essential for determining bases and dimensions of subspaces.

8. Orthogonal Subspaces: Subspaces can be orthogonal, meaning any vector in one subspace is orthogonal to any vector in the other.

9. Applications: Linear subspaces are foundational in areas such as linear algebra, functional analysis, and more applied fields like machine learning and computer graphics.

Review these points to grasp the essential characteristics and properties of linear subspaces in vector spaces.

When studying the "Basis of a subspace," focus on the following key points:

1. Definition of Subspace: A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication.

2. Span of a Set of Vectors: The span of a set of vectors is the collection of all linear combinations of those vectors. A subspace can be defined as the span of some vectors.

3. Basis: A basis for a subspace is a set of vectors that:

   • Span the subspace.
   • Are linearly independent (no vector in the set can be written as a linear combination of the others).

4. Dimensions: The dimension of a subspace is the number of vectors in its basis. This represents the number of degrees of freedom within the subspace.

5. Finding a Basis: To find a basis:

   • Identify a spanning set for the subspace.
   • Use techniques such as row reduction (Gaussian elimination) to determine linear independence and reduce the spanning set to a basis.

6. Properties: Any two bases of a subspace have the same number of vectors (dimension) and can be transformed into each other through linear combinations.

7. Orthogonal Basis: A special type of basis where vectors are orthogonal (perpendicular) to each other, often simplified using the Gram-Schmidt process.

8. Applications: Understanding bases is crucial in various fields, including computer science, physics, engineering, and data analysis.

By mastering these key points, you will grasp the fundamental concepts related to the basis of a subspace in linear algebra.