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 $W$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.