Independent Subspaces

Definition

Two Subspace $S_1,S_2$ are independent if: $S_1\cap S_2=\{0\}$

Alternate Definitions

• Let $W_1,\dots ,W_k$ be subspaces of $V$. $W_1,\dots ,W_k$ are said to be independent if ${\sum }_{i=1}^k{\alpha }_i=0$, where ${\alpha }_i\in W_i$, implying that ${\alpha }_j=0,\forall j$.
• $\forall j\in \{1,\dots ,k\},W_j\cap \{W_1,\dots ,W_{j-1}\}=0$
• If ${\beta }_1,\dots ,{\beta }_k$ are Ordered Basis for $W_1,\dots ,W_k$, Then $\beta =\{{\beta }_1,\dots ,{\beta }_k\}$ is an ordered basis for $W=W_1+\cdots +W_k$. The final sum is the Direct Sum $W=W_1\oplus \cdots \oplus W_k$

Intuition

The only shared factor between them is the trivial linear combination.