Quotients and direct sums

Linear Algebra II studies a single operator $T: V\to V$ in depth. The first tools are ways of splitting and collapsing spaces.

1. Direct sums (Def. 1.1, Prop. 1.2) and complements (Prop. 1.3).
2. Projections (Def. 1.4) correspond exactly to direct sum decompositions (Prop. 1.5).
3. Quotient spaces (Def. 1.6, Prop. 1.7) and the first isomorphism theorem (Thm 1.8), which refines rank–nullity.

For subspaces $U_1, \dots, U_k$ of $V$, the sum $U_1 + \dots + U_k = \{u_1 + \dots + u_k : u_i\in U_i\}$ is direct, written $U_1\oplus\dots\oplus U_k$, if every vector in it can be written as $u_1 + \dots + u_k$ ($u_i\in U_i$) in exactly one way.

For subspaces $U_1, \dots, U_k$ of a finite-dimensional $V$, the following are equivalent:

1. the sum is direct;
2. $u_1 + \dots + u_k = 0$ with $u_i\in U_i$ forces every $u_i = 0$;
3. $\dim(U_1 + \dots + U_k) = \dim U_1 + \dots + \dim U_k$;
4. concatenating bases of the $U_i$ gives a basis of the sum. For two subspaces, these are also equivalent to $U_1\cap U_2 = \{0\}$.

If $U$ is a subspace of finite-dimensional $V$ then there is a subspace $W$ (a complement) with $V = U\oplus W$. Complements are not unique unless $U = 0$ or $U = V$.

A linear map $P: V\to V$ is a projection if $P^2 = P$. If $V = U\oplus W$, the projection onto $U$ along $W$ sends $u + w\mapsto u$.

If $P^2 = P$ then $V = \operatorname{im}P\oplus\ker P$, $P$ is the identity on $\operatorname{im}P$, and $P$ is the projection onto $\operatorname{im}P$ along $\ker P$. Also $I - P$ is a projection, onto $\ker P$ along $\operatorname{im}P$. Conversely, the projection along any decomposition $V = U\oplus W$ satisfies $P^2 = P$ with image $U$ and kernel $W$. So projections correspond exactly to ordered pairs of complementary subspaces.

For a subspace $U\le V$, the coset of $v$ is $v + U = \{v + u : u\in U\}$; $v + U = v' + U$ iff $v - v'\in U$. The quotient space $V/U$ is the set of cosets with

and the quotient map $\pi: V\to V/U$, $v\mapsto v + U$.

The operations on $V/U$ are well defined and make $V/U$ a vector space; $\pi$ is linear and surjective with kernel $U$. If $u_1, \dots, u_k, w_1, \dots, w_m$ is a basis of $V$ extending a basis of $U$, then $w_1 + U, \dots, w_m + U$ is a basis of $V/U$. So $\dim V/U = \dim V - \dim U$.

If $T: V\to W$ is linear, then $\bar T: V/\ker T\to\operatorname{im}T$, $v + \ker T\mapsto Tv$, is a well-defined isomorphism. More generally, if $U\le\ker T$ then $\bar T: V/U\to W$, $v + U\mapsto Tv$ is well defined and linear with $T = \bar T\circ\pi$.

Let $T: V\to V$ and $U\le V$ with $T(U)\subseteq U$. Then $T$ restricts to $T|_U: U\to U$ and induces $\bar T: V/U\to V/U$, $v + U\mapsto Tv + U$. In a basis $u_1, \dots, u_k, w_1, \dots, w_m$ extending a basis of $U$,

so $\chi_T = \chi_{T|_U}\,\chi_{\bar T}$.

• Direct sum of two: check $U\cap W = 0$ and $\dim U + \dim W = \dim V$.
• Three or more: check dimensions add; pairwise intersections are not enough.
• Projection: check $P^2 = P$; read off image and kernel.
• Quotient basis: extend a basis of $U$; use the cosets of the extra vectors.
• Maps from $V/U$: define on representatives, then check $U$ maps to $0$.