Let V be a vector space. A subset of the form A={𝐰+𝐚 |𝐰 ∈W}, where W ⊂V is a subspace and 𝐚 ∈V i

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Let V be a vector space. A subset of the form A={𝐰+𝐚 |𝐰 ∈W},...

Let $V$ be a vector space. A subset of the form $A=\{\mathbf{w}+\mathbf{a} \mid \mathbf{w} \in W\}$, where $W \subset V$ is a subspace and $\mathbf{a} \in V$ is a fixed vector, is known as an affine subspace of $V$. (a) Show that an affine subspace $A \subset V$ is a genuine subspace if and only if $\mathbf{a} \in W$. (b) Draw the affine subspaces $A \subset \mathbb{R}^2$ when (i) $W$ is the $x$-axis and $\mathbf{a}=(2,1)^T$, (ii) $W$ is the line $y=\frac{3}{2} x$ and $\mathbf{a}=(1,1)^T$, (iii) $W$ is the line $\left\{(t,-t)^T \mid t \in \mathbb{R}\right\}$, and $\mathbf{a}=(2,-2)^T$. (c) Prove that every affine subspace $A \subset \mathbb{R}^2$ is either a point, a line, or all of $\mathbb{R}^2$. (d) Show that the plane $x-2 y+3 z=1$ is an affine subspace of $\mathbb{R}^3$. (e) Show that the set of all polynomials such that $p(0)=1$ is an affine subspace of $\mathcal{P}^{(n)}$.

Key Concepts

Polynomial Vector Space

A polynomial vector space consists of polynomials of a certain degree or less, which can be added together and multiplied by scalars, satisfying the axioms of a vector space. When considering subsets of these spaces defined by a condition such as the value at a specific point, these subsets often form affine subspaces because they can be seen as translations of subspaces that satisfy homogeneous conditions.

Vector Space

A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars, and these operations satisfy properties such as associativity, distributivity, and the existence of an additive identity and additive inverses. This structure forms the foundational setting for linear algebra, enabling the study of linear combinations, span, basis, and dimension.

Subspace

A subspace is a subset of a vector space that is itself a vector space, meaning it must be closed under vector addition and scalar multiplication, and it must contain the zero vector. Subspaces enable us to analyze parts of a vector space that retain the full algebraic structure, and they often serve as the building blocks for more complex constructions like direct sums and quotient spaces.

Affine Subspace

An affine subspace is a translation of a subspace by a fixed vector. Essentially, it is formed by taking all vectors in a subspace and adding a constant vector to each one, creating a set that may not pass through the origin. This concept preserves many linear properties but does not satisfy the conditions to be a subspace unless the translation vector is the zero vector.

Translation

Translation in the context of vector spaces refers to the operation of shifting every element of a subspace by a fixed vector. This process produces an affine subspace and is central to distinguishing between linear subspaces (which contain the origin) and affine subspaces (which do not necessarily do so), highlighting the idea of a coset.

Dimension and Classification

The dimension of a subspace indicates the number of vectors in a basis for that subspace and provides a measure of its size. In the context of affine subspaces, the dimension of the associated subspace dictates whether the affine subspace is just a point (dimension zero), a line (dimension one), or a higher-dimensional flat like a plane. This classification is particularly evident in two-dimensional space, where affine subspaces can only be points, lines, or the whole space.

Affine Hyperplane

An affine hyperplane is a specific type of affine subspace that has one dimension less than the ambient space. It can often be defined by a linear equation, and while it is not a subspace if it does not pass through the origin, it retains a flat, affine structure. This concept is particularly useful in understanding geometric configurations in higher-dimensional spaces.

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