Linear Algebra MT23, Quotient spaces
Flashcards
What is meant by a coset of $\mathcal U$ in $V$, i.e. $v + \mathcal U$?
The translates of $\mathcal U$, given by
\[\{v + u \mid u \in \mathcal U\}\]Let $V$ be a (possibly infinitely dimensional) vector space. Let $U \le V$. @Define $V/U$.
with
\[(v + U) + (v' + U) = (v + v') + U\]and
\[\lambda(v + U) = (\lambda v) + U\]What useful fact lets you decide whether $v _ 1 + U = v _ 2 + U$?
Suppose $e _ 1, \ldots, e _ k$ is a basis for $U \le V$ and $e _ 1, \ldots, e _ k, e _ {k+1}, \ldots, e _ n$ is a basis for $V$. Then can you give a basis for $V/U$?
Suppose $V$ is a finite dimensional vector space and $U \le V$. What is
\[\dim V/U\]
?
@State the 1st isomorphism theorem for vector spaces.
Let $T : V \to W$ be linear. Then
\[V/\ker T \cong \mathrm{Im}\, T\]via
\[v + \ker T \mapsto Tv\]When does a map $T : V \to W$ descend to a well defined map of quotients $\overline T : V/A \to W/B$, given by
\[\overline T(v + A) = T(v) + B\]
?
When
\[T(A) \subseteq B\]Proofs
Prove that a map $T : V \to W$ descend to a well defined map of quotients $\overline T : V/A \to W/B$, given by
\[\overline T(v + A) = T(v) + B\]
if and only if
\[T(A) \subseteq B\]
The $T(A) \subseteq B$ direction is okay, just work through the definitions with different representatives $v$ and $v'$ for $v + A$.
For the other direction, assume that $\overline T$ is well defined and let $a \in A$. We want to show that $T(a) \in B$. Then note that
\[\begin{aligned} B &= 0 _ {W/B} + B \\ &= \overline T(0 _ {V/A}) \\ &= \overline T(A) \\ &= \overline T(a + A) \\ &= T(a) + B \end{aligned}\]so $T(a) \in B$. Then $T(A) \subseteq B$. (Above, $0 _ {W/B}$ and $0 _ {V/A}$ denote the $0$ elements of the corresponding vector spaces).