In Linear Algebra Done Right, the book proves that every subspace of $V$ is part of a direct sum equal to $V$. I generally follow the proof, but do not understand some points.
Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. Then there is a subspace $W$ of $V$ such that $V = U\bigoplus W$.
Proof
Because $V$ is finite-dimensional, so is $U$ (proved in 2.26. I am OK). Thus there is a basis $u_1,…,u_m$ of $U$. Of course $u_1,…u_m$ is a linearly independent list of vectors in $V$ (I am OK). Hence this list can be extended to a basis $u_1,…,u_m, w_1,…,w_n$ of $V$ (I am not OK. Does it mean $w_1,…w_n$ can only be vectors that extend to the basis of $V$? If so, why the title of the proof said EVERY subspace of $V$ is part of a direct sum equal to $V$).
To prove $V = U\bigoplus W$, we need only show that $$V = U+W \text{ and } U \cap W = \{0\}$$ (I am OK).
To prove the first equation above, suppose $v \in V$. Then, because the list $u_1,…,u_m, w_1,…w_n$ spans $V$, there exist $a_1,…a_m, b_1,…b_m \in \mathbb{F}$ such that $$v=a_1u_1+…+a_mu_m+b_1w_1+…+b_nw_n$$.
In other words, we have $v=u+w$, wheere $u \in U$ and $w \in W$ are defined as above. Thus $v \in U + W$, completing the proof that $V = U + W$. (I am also OK.)
To show that $U \cap W = \{0\}$, suppose $v \in U \cap W$. Then there exist scalars $a_1,…,a_m,b_1,…b_n \in \mathbb{F}$ such that
$$v=a_1v_1+…+a_mv_m = b_1w_1+…+b_nw_n$$. Thus
$$a_1u_ + …+a_mu_m – b_1w_1-…-b_nw_n = 0$$
Because $u_1,…u_m,w_1,…,w_n$ is linearly independent, this implies that $a_1=…=a_m=b_1=…=b_n = 0$. Thus $v=0$, completing the proof that $U \cap W = \{0\}$.
Is the proof means for every subspace $U$ of finite-dimensional $V$, we can find a $W$ that is the direct sum of $V$?
Best Answer
You have a subspace $U$ with basis $u_1,...,u_m$.
Extend the basis to a basis of $V$ by adding vectors $w_1,...,w_n$. There is some freedom to choose the $w_k$ but they must be linearly indepdendent and the collection must span $V$.
Let $W=\operatorname{sp} \{w_k\}$.
Since the whole collection spans $V$, we must have $V = U +W$. If $u\in U, w\in W$ and $u+w = 0$, we must have $u=w=0$ since the whole collection is linearly independent.
Note that $W$ is not unique.