Exercise 2B Problem 8

Problem

Suppose $U$ and $W$ are subspaces of $V$ such that $V = U \oplus W$. Suppose also that $u_1, \dots, u_m$ is a basis of $U$ and $w_1, \dots, w_n$ is a basis of $W$. Prove that $$u_1, \dots, u_m, w_1, \dots, w_n$$ is a basis of $V$

Solution

Every vector in $V$ can be written uniquely as a vector $u \in U$ plus a vector $w \in W$ as per the fact that $V = U \oplus W$. Therefore if $v \in V$, then for a specific $u \in U$ and $w \in W$ we have $v = u + w$.

Since $u_1, \dots, u_m$ is a basis for $U$ any vector $u \in U$ can be written uniquely in terms of this basis as per theorem 2.29 and similarly for any $w \in W$. Therefore $v = u + w$ is uniquely determined in the list $u_1, \dots, u_m, w_1, \dots, w_n$ which by theorem 2.29 implies that $u_1, \dots, u_m, w_1, \dots, w_n$ is a basis for $V$.