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Given $V$ vector space, and $W$ a subspace of $V$. How can I get the equations that describe $W$ if I'm only given its basis?

I would like to know the logic behind it rather than just the procedure or the direct answer

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    Given a basis $b_1,...,b_d$ then $W = \{ \sum_k x_k b_k \}_x$ where the $x_k$ take any scalar values. – 2017-02-25
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    Yes, but what I want to know its for example if W $\in$ $\mathbb{R}^4$ I have equations like $x+y-2z=0$. How can I find those ones if I'm only given the basis of the subspaces – 2017-02-25

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The main idea is that $W$ is described by saying it's precisely the set of vectors orthogonal to $W^\perp$, the orthogonal complement. We find a basis for $W^\perp$ by solving $Ax=0$, where the basis vectors for $W$ are the row vectors of $A$. [To say $x$ solves $Ax=0$ is to say precisely that $x$ is orthogonal to the rows of $A$, and hence to any linear combination of them, i.e., orthogonal to $W$.] Then: If $v_1,\dots,v_k$ is a basis for $W^\perp$, i.e., for the nullspace of $A$, then $W$ is given precisely by the equations $v_1\cdot x = v_2\cdot x = \dots = v_k\cdot x = 0$. (The fancy way of writing this is to say that $W=(W^\perp)^\perp$.)

[It might help to watch Lecture 38 — or perhaps more — of these lectures.]