23 - Direct Sum

Direct Sum

Let $U,W$ be subspaces of a vector space $V$

If $U\cap W=\{0_V\}$ and $U+W=V$
Then

$$V\text{ is the direct sum of }U\text{ and }W⟹V=U\oplus W$$

Equivalent Statement for Direct Sums

Let $U,W$ be subspaces of a finite-dimensional vector space $V$

1. $V=U\oplus W$
2. Every $v\in V$ has a unique expression as $u+w$ for $u\in U$ and $w\in W$
3. $\dim V=\dim U+\dim W$ and ${} V = U + W
4. $\dim V=\dim U+\dim W$ and $U\cap W=\{0_V\}$
5. If $u_1,\cdots ,u_m$ is a basis for $U$ and $w_1,\cdots ,w_n$ is a basis for $W$ then

$$u_1,\cdots ,u_m,w_1,\cdots ,w_n\text{ is a basis for }V$$

Proof

TODO

Direct Sums on multiple Subspaces / Vector Spaces

1. Internal Direct Sum
   Vector space $V$ is a direct sum of sum of subspaces $X_1,\cdots ,X_k\le V$
   If every $v\in V$ can be uniquely written

$$v=x_1+x_2+\cdots +x_k\text{ where }{x}_i\in X_i\text{ for all }i$$

The general expression is the general form of (2) from the above

2. External Direct Sum
   Given vector spaces $V_1,\cdots ,V_k$ then the external direct sum

$$V_1\oplus V_2\oplus \cdots \oplus V_k$$

has the Cartesian product $V_1\times V_2\times \cdots \times V_k$ as the underlying set with
Addition and Scalar Multiplication defined component wise with

$$\begin{array}{rl}(v_1,\cdots ,v_k)+(w_1,\cdots ,w_k)& =(v_1+w_1,\cdots ,v_k+w_k)\\ \alpha \cdot (v_1,\cdots ,v_k)& =(\alpha \cdot v_1,\cdots ,\alpha \cdot v_k)\end{array}$$