Exact sequences compress kernel-image bookkeeping into diagrams. They are a language for saying that one linear construction loses exactly the information that the next one receives.
Exactness
Definition. A sequence of vector spaces and linear maps \[ \cdots\to V_{i-1}\xrightarrow{T_{i-1}}V_i\xrightarrow{T_i}V_{i+1}\to\cdots \] is exact at \(V_i\) if \(\operatorname{im}T_{i-1}=\ker T_i\). It is exact if it is exact at every displayed middle term.
Example. For any linear map \(T:V\to W\), the sequence \[ 0\to\ker T\to V\xrightarrow{T}\operatorname{im}T\to0 \] is exact. This is the rank-nullity theorem written as a diagram.
Short exact sequences
Definition. A short exact sequence is an exact sequence \[ 0\to U\xrightarrow{i}V\xrightarrow{p}W\to0. \] This means \(i\) is injective, \(p\) is surjective, and \(\operatorname{im}i=\ker p\).
Theorem. If \[ 0\to U\to V\to W\to0 \] is a short exact sequence of finite-dimensional vector spaces, then \[ \dim V=\dim U+ \dim W. \]
Proof. Identify \(U\) with its image in \(V\). The quotient map \(V\to W\) has kernel \(U\), so by the first isomorphism theorem \(V/U\cong W\). Therefore \(\dim W=\dim V- \dim U\).
Splitting
Theorem. Every short exact sequence of vector spaces splits: if \[ 0\to U\xrightarrow{i}V\xrightarrow{p}W\to0 \] is exact, then there exists a linear map \(s:W\to V\) with \(p\circ s=I_W\), and \[ V\cong U\oplus W. \]
Proof. Choose a basis \((w_j)\) of \(W\). Since \(p\) is surjective, choose \(v_j\in V\) with \(p(v_j)=w_j\), and define \(s(w_j)=v_j\). Then \(p\circ s=I_W\). Every \(v\in V\) decomposes as \((v-sp(v))+sp(v)\), with the first term in \(\ker p=\operatorname{im}i\).
Commutative diagrams
Proposition. A diagram commutes when all paths with the same start and end give the same linear map. For example, saying \[ \begin{array}{ccc} V&\xrightarrow{T}&W\\ \downarrow &&\downarrow\\ V'&\xrightarrow{T'}&W' \end{array} \] commutes means \(aT=T'b\), where \(b:V\to V'\) and \(a:W\to W'\) are the vertical maps.
Dimension from long exact chains
Proposition. If \[ 0\to V_0\to V_1\to\cdots\to V_n\to0 \] is exact and all spaces are finite-dimensional, then the alternating sum of dimensions is zero: \[ \sum_{i=0}^n(-1)^i\dim V_i=0. \]
Remark. Exact sequences are not extra abstraction for its own sake. They prevent mistakes by forcing every lost dimension to appear as a kernel and every visible dimension to appear as an image.