Subspaces and Quotients

A vector space can contain smaller vector spaces called subspaces. We can also “divide” a space by a subspace to create a quotient space.

Subspaces

A subset $W$ of a vector space $V$ is a subspace if:

1. The zero vector $\mathbf{0} \in W$.
2. $W$ is closed under addition: $\mathbf{u}, \mathbf{v} \in W \implies \mathbf{u} + \mathbf{v} \in W$.
3. $W$ is closed under scalar multiplication: $c \in F, \mathbf{v} \in W \implies c\mathbf{v} \in W$.

Quotient Spaces

Given a subspace $W \subseteq V$, the quotient space $V/W$ is the set of all cosets $\mathbf{v} + W$. Intuitively, the quotient space “ignores” all differences that lie within $W$.

The dimension of a quotient space is: $\dim(V/W) = \dim(V) - \dim(W)$

Kernels and Images

For a linear map $T: V \to W$:

• The Kernel $\ker(T) = \{ \mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0} \}$ is a subspace of $V$.
• The Image $im(T) = \{ T(\mathbf{v}) \mid \mathbf{v} \in V \}$ is a subspace of $W$.