A set is a subspace if it satisfies the following two conditions: 1. It contains the zero vector (the additive identity). 2. It is closed under vector addition and scalar multiplication. In other words, a set is a subspace if it is closed under the operations of vector addition and scalar multiplication, and contains the zero vector. This means that any linear combination of vectors in the set must also be in the set, and the set must be closed under addition and scalar multiplication. **

Affine subspaces are sets of points in a vector space that are closed under affine combinations. An affine combination of points is a weighted sum of the points where the weights sum to 1. Affine subspaces can be thought of as generalizations of lines, planes, and hyperplanes in higher dimensions. They can be represented as translations of linear subspaces, and they are characterized by the property that any two points in the subspace determine a unique line contained in the subspace. **

Subspaces of mappings refer to the spaces that are formed by the collection of all possible outputs of a given mapping function. These subspaces are subsets of the codomain of the mapping and can include vectors, functions, or other mathematical objects. By studying these subspaces, mathematicians can gain insight into the properties and behavior of the mapping function, helping to analyze its structure and relationships with other mathematical objects. Understanding subspaces of mappings is crucial in various fields of mathematics, such as linear algebra, functional analysis, and differential equations. **

Subspaces are a fundamental concept in linear algebra and refer to a subset of a vector space that is itself a vector space. In other words, a subspace is a set of vectors that is closed under addition and scalar multiplication. The dimension of a subspace is the number of linearly independent vectors needed to span that subspace. It represents the "size" of the subspace and provides important information about the structure of the vector space. The dimension of a subspace is always less than or equal to the dimension of the original vector space. **

To determine if a subset is a subspace, it must satisfy three conditions: it must contain the zero vector, it must be closed under vector addition, and it must be closed under scalar multiplication. If a subset satisfies all three conditions, then it is a subspace. **

These are the only subspaces of R2 because any subspace of R2 must satisfy two properties: it must contain the zero vector, and it must be closed under vector addition and scalar multiplication. The subspaces listed (the zero subspace, the x-axis, the y-axis, and R2 itself) all satisfy these properties. Any other set of vectors in R2 would not satisfy these properties and therefore would not be a subspace. **

To determine the dimension of a subspace, you can find a basis for the subspace and count the number of vectors in the basis. The number of vectors in the basis will give you the dimension of the subspace. Alternatively, you can use the rank-nullity theorem, which states that the dimension of a subspace is equal to the sum of the dimensions of its intersection with the null space and its orthogonal complement. By finding the dimensions of these two spaces, you can determine the dimension of the subspace. **

The set of all vectors of the form (x, 0) where x is a real number is a subspace of ℝ² because it contains the zero vector, is closed under addition, and is closed under scalar multiplication. The set of all vectors of the form (0, y) where y is a real number is also a subspace of ℝ² for the same reasons. However, the set of all vectors of the form (x, x) where x is a real number is not a subspace of ℝ² because it does not contain the zero vector. **