Affine Subspaces

Let $V$ be a vector space.

<aside> 💡 An affine subspace of $V$ is a subset $L$ such that there exists $v\in V$ and a vector subspace $U\subseteq V$ such that: $L=v+U:=\{ v+u|u \in U\}$

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• We say an affine subspace $L$ is $n$-dimensional if the corresponding $U$ is $n$-dimensional.
• The subspace $U$ corresponding to an affine subspace is unique.
• Affine subspaces are thus translates of a vector subspace of $V.$

Affine Subspaces in $\R^2$

• Points
• Lines
• The entire plane $\R^2$.

Affine Subspaces in $\R^3$

• Points

• Lines

• Planes

• The entire space $\R^3$

  

The Solution Set to a System of Linear Equations