Direct sum

The sum of two subspaces is the subspace $U+V=\text{span}(U \cup V)=\{u+v \in W\:|\:u \in U \wedge v \in V\}$
If $U \cap V=\{\boldsymbol{0}\}$, then the sum is called the direct sum and is denoted by $U \oplus V$.

Put differently, $W=U\oplus V$ iff for every $w \in W$ there exist unique vectors $u \in U$ and $v \in V$ such that $w=u+v$.

If $U_i \leq W$ for $1 \leq i \leq k$, then $W=U_1\oplus \cdots \oplus U_k$ iff $W=U_1+U_2+\cdots+U_k$ and $U_r \cap \sum\limits_{i \neq r} U_i=\{0\}$ for $1 \leq r \leq k$.
Note: It is not sufficient that $U_i \cap U_j=\{0\}$ whenever $i \neq j$.

Examples:
The complex numbers are the direct sum of the real and purely imaginary numbers. $\mathbb{C}=\mathbb{R}\oplus i\mathbb{R}$

$\langle e_1 \rangle+\langle e_2,e_3 \rangle=\langle e_1 \rangle \oplus \langle e_2,e_3 \rangle$ is a direct sum but $\langle e_1,e_2 \rangle+\langle e_2,e_3 \rangle$ is not a direct sum since $\langle e_1,e_2 \rangle \cap \langle e_2,e_3 \rangle= \langle e_2 \rangle \neq \{\boldsymbol{0} \}$.

Any function is a sum of even and odd function.
$E=\{f \in F(\mathbb{R})|f(t)=f(-t)\},O=\{f \in F(\mathbb{R})|f(t)=-f(-t)\}$
$E\oplus O=F(\mathbb{R})$
$f(t)=\frac{1}{2}[f(t)+f(-t)-f(-t)+f(t)]=\frac{1}{2}[f(t)+f(-t)]+\frac{1}{2}[f(t)-f(-t)]$

Denote the vector space of square matrices by $M_{n \times n}(\mathbb{R})$.
Denote by $M_{n \times n}(\mathbb{R})_+, M_{n \times n}(\mathbb{R})_-$ the vector subspaces of symmetric and skew-symmetric matrices.
$M_{n \times n}(\mathbb{R})=M_{n \times n}(\mathbb{R})_+\oplus M_{n \times n}(\mathbb{R})_-$
Let $X \in M_{n \times n}(\mathbb{R})$. Then $X_+=\frac{1}{2}(X+X^T)$ and $X_-=\frac{1}{2}(X-X^T)$.
$X=\frac{1}{2}(X+X^T)+\frac{1}{2}(X-X^T)$

Complex square matrices is the direct sum of Hermitian and skew-Hermitian matrices.
Denote the vector space of complex square matrices by $M_{n \times n}(\mathbb{C})$.
Denote by $M_{n \times n}(\mathbb{C})_+, M_{n \times n}(\mathbb{C})_-$ the vector subspaces of Hermitian and skew-Hermitian matrices.
$M_{n \times n}(\mathbb{C})=M_{n \times n}(\mathbb{C})_+\oplus M_{n \times n}(\mathbb{C})_-$
Let $A \in M_{n \times n}(\mathbb{C})$. Then $A_+=\frac{1}{2}(A+A^*)$ and $A_-=\frac{1}{2}(A-A^*)$.
$A=\frac{1}{2}(A+A^*)+\frac{1}{2}(A-A^*)$

Important idea:
If $U \cap V=\{ \boldsymbol{0} \}$ and $U, V$ are subspaces of a vector space with bases $\{u_1, \cdots, u_k\}$ and $\{v_1, \cdots, v_l\}$ respectively, then $\{u_1, \cdots, u_k, v_1, \cdots, v_l \}$ is a basis for $U \oplus V$. Therefore, knowing that $X \oplus Y=X+Y$ allows us to find a basis of $X+Y$ easily: we just find the union of bases of $X$ and $Y$. Namely, if $V=U_1 \oplus U_2 \oplus \cdots \oplus U_k$ and $B_i$ is a basis of $U_i$ then $B_1 \cup B_2 \cup \cdots \cup B_k$ is a basis of V. In particular, $\text{dim}\:V=\sum\limits_{i=1}^k \text{dim}\: U_i$.

Direct sum of vectors
Any vector $\vec{v}\in V$ can be written as a linear combination of the basis vectors, $\vec{v}=v_1e_1+\cdots+v_ne_n$. We normally express this fact in the form of a column vector, $\vec{v}=(v_1,v_2,\cdots,v_n)^T$. Similarly, for any vector $\vec{w}$, we can write $\vec{w}=(w_1,w_2,\cdots,w_m)^T$. Thus, $\vec{v}=(v_1,v_2,\cdots,v_n,0,0,\cdots,0)^T$ and $\vec{w}=(0,0,\cdots,0,w_1,w_2,\cdots,w_m)^T$. Then a direct sum of two non-zero vectors is $\vec{v}\oplus \vec{w}=(v_1,v_2,\cdots,v_n,w_1,w_2,\cdots,w_m)^T$.

Block-diagonal form
A matrix is actually a linear map from a vector space to another vector space. Let's consider the endomorphism $A:V\to V, \vec{v}\mapsto A\vec{v}$ or $\begin{pmatrix} v_1\\v_2\\ \vdots\\ v_n\end{pmatrix}\mapsto \begin{pmatrix} a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}\begin{pmatrix} v_1\\v_2\\ \vdots\\ v_n\end{pmatrix}.$ Similarly, $B:W\to W, \vec{w}\mapsto B\vec{w}$. On the direct sum space $V\oplus W$, the matrix $A\oplus B$ still acts on the vectors such that $\vec{v}\mapsto A\vec{v}$ and $\vec{w}\mapsto B\vec{w}$. This is achieved by lining up all matrix elements in a block-diagonal form, $$A\oplus B=\begin{pmatrix}A&0_{n\times m}\\ 0_{m\times n}&B\end{pmatrix}.$$
For example, if $n=2,m=3$, and $A=\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix},B=\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\\b_{31}&b_{32}&b_{33}\end{pmatrix}$, then $$A\oplus B=\begin{pmatrix}a_{11}&a_{12}&0&0&0\\a_{21}&a_{22}&0&0&0\\0&0&b_{11}&b_{12}&b_{13}\\0&0&b_{21}&b_{22}&b_{23}\\0&0&b_{31}&b_{32}&b_{33}\end{pmatrix}.$$
When you act it on $\vec{v}\oplus \vec{w}$, $$(A\oplus B)(\vec{v}\oplus \vec{w})=\begin{pmatrix}A&0_{n\times m}\\ 0_{m\times n}&B\end{pmatrix}\begin{pmatrix}\vec{v}\\ \vec{w}\end{pmatrix}=\begin{pmatrix}A\vec{v}\\B\vec{w}\end{pmatrix}=(A\vec{v})\oplus(B\vec{w}).$$ If you have two matrices, their multiplications are done on each vector space separately, $$(A_1 \oplus B_1)(A_2\oplus B_2)=(A_1A_2)\oplus (B_1B_2).$$
Note that not every matrix on $V\oplus W$ can be written as a direct sum of a matrix on $V$ and another on $W$. There are $(n+m)^2$ independent matrices on $V\oplus W$, while there are only $n^2$ and $m^2$ matrices on $V$ and $W$, respectively. Remaining $(n+m)^2-n^2-m^2=2mn$ matrices cannot be written as a direct sum.

Other useful formulae are $\text{det}(A\oplus B)=\text{det}(A)\text{det}(B)$ and $\text{Tr}(A\oplus B)=\text{Tr}(A)+\text{Tr}(B)$.

More to learn:
geometric picture of sum of two subspaces and examples
http://algebra.math.ust.hk/vector_space/13_direct_sum/lecture2.shtml

more on direct sum
https://math.dartmouth.edu/archive/m22s02/public_html/VectorSpaces.pdf

examples
http://mathwiki.ucdavis.edu/Algebra/Linear_algebra/04._Vector_spaces/4.4_Sums_and_direct_sum

projection operators
https://people.maths.ox.ac.uk/flynn/genus2/alg0506/LALect02.pdf

problems on direct sum
http://en.wikibooks.org/wiki/Linear_Algebra/Combining_Subspaces

http://math.stackexchange.com/questions/1151335/what-is-direct-sum-decomposition
https://web.math.princeton.edu/~jonfick/2012SpMAT204/Direct_Sums.pdf
http://www.math.nyu.edu/faculty/hausner/quotientspaces.pdf
http://www.math.ttu.edu/~drager/Classes/05Summer/m5399/struct.pdf
http://www.mathsman.co.uk/LinearAlgebra-Ver1.4[1].pdf
http://planetmath.org/directsumofmatrices