To expand slightly on  Minhyong's comment, the key facts can be
found in Fulton's Intersection Theory. If you look  at the comment  following
corollary 18.3.2, you'll see an isomorphism (in slightly different notation)
$$ch:K^0(X)\otimes \mathbb{Q}\cong CH(X)\otimes\mathbb{Q}$$
where $X$ is a nonsingular variety, $K^0(X)$ is the Grothendieck group of
vector bundles, $CH(X)$ is the Chow group of cycles mod rational equivalence,
and $ch$ is the Chern character. After mapping this to rational cohomology, you get exactly
the statement you want.

Note that this is false 

 1. if you omit the $\mathbb{Q}$, or
 2. if you work on a general (compact) complex manifolds because there may not be enough vector bundles. (To be clear, I mean that conclusion in cohomology is false. For
$X$ take a general torus.)