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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 $\mathbb{C}^5$ ?

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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 $W$

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