# Extend basis of a subspace to a basis of $\mathbb{C}^5$

Let $$U$$ be the subspace of $$\mathbb{C}^5$$ defined by
$$U = \{(z_1, z_2, z_3, z_4, z_5) \in \mathbb{C}^5: 6z_1 = z_2\;\wedge\; z_3 + 2z_4 + 3z_5 =0 \}$$

(a) Find a basis of $$U$$

(b) Extend the basis in part(a) to a basis of $$\mathbb{C}^5$$.

(c) Find a subspace $$W$$ of $$\mathbb{C}^5$$ such that $$\mathbb{C}^5 = U \oplus W$$

For part (a), by using two conditions, I find the basis of $$U$$ is $$\{(1,6,0,0,0), (0,0,-2,1,0), (0,0,-3,0,1)\}$$

But how can I extend it to a basis of $$\mathbb{C}^5$$ ?

Consider any spanning set, for example, the standard vectors $$\{e_1,e_2,e_3,e_4,e_5\}$$. Call your three basis vectors as $$v_1,v_2,v_3$$. If $$e_1$$ is a linear combination of $$v_i$$'s then do not add $$e_1$$ to your list. If this is not the case , then add $$e_1$$ to the list and so now it has four independent vectors. Can you continue?
For the next part, we already have a basis $$\{v_1,v_2,v_3,e_k,e_m\}$$ of $$\Bbb C^5$$ where $$e_k$$ and $$e_m$$ are standard vectors obtaining in the previous part. Now it is easy to check that $$\text{span}(e_k,e_m)$$ is the required $$W$$