Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the product of the rational cohomology groups $H^{4j}(K)$ is rationally an isomorphism.
(2). Therefore in order for a vector bundle $\xi$ on $K$ to be such that some multiple of it is trivial it is sufficient if the rational Pontryagin classes vanish.