# Notes - Linear Algebra MT23, Quotient spaces

> Source: https://ollybritton.com/notes/uni/part-a/mt23/linear-algebra/notes/quotient-spaces/ · Updated: 2023-10-19 · Tags: uni, notes

- [Course - Linear Algebra MT23](https://ollybritton.com/notes/uni/part-a/mt23/linear-algebra/)

### 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 (possibily infinitely dimensional) vector space. Let $U \le V$. Can you define $V/U$?::
$$
V/U = \\{v + U \mid v \in V\\}
$$
with
$$
(v + U) + (v' + U) = (v + v') + U
$$
and
$$
\lambda(v + U) = (\lambda v) + U
$$

What useful fact lets you decide whether $v_1 + U = v_2 + U$?::
$$
v_1 - v_2 \in U
$$

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 $V/U$?::
$$
e_{k+1} + U, \ldots, e_n + U
$$

Suppose $V$ is a finite dimensional vector space and $U \le V$. What is
$$
\dim V/U
$$
?::
$$
\dim V - \dim U
$$

Can you state the 1st isomorphism theorem for vector spaces?::
Let $T : V \to  W$ be linear. Then
$$
V/\ker T \cong \text{Im } T
$$
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
$$
\overline T(v + A) = T(w) + B
$$
?::
When
$$
T(A) \subseteq B
$$

### Proofs
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 $V$. Prove that
$$
e_{k+1} + U, \ldots, e_n + U
$$
is a basis for $V/U$.::
Todo.

Prove the 1st isomorphism theorem for vector spaces, i.e. let $T : V \to  W$ be linear. Then
$$
V/\ker T \cong \text{Im } T
$$
via
$$
v + \ker T \mapsto Tv
$$
Furthermore, derive the rank-nullity theorem as a corollary.::
Todo.

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
$$
\overline T(v + A) = T(w) + B
$$
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).

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