Given a vector space $V=\mathbb{F}^{d}$, the free algebra, or tensor algebra, of $V$ is $T\left(V\right)=\oplus_{n\geq0}V^{\otimes n}$. Now, it is stated everywhere, that this is exactly the algebra of non-commutative polynomials over $d$ indeterminates.
I'm probably missing something really basic, but can someone please show me why this is true? I would really appreciate an intuitive explanation with a fromal one, as i'm having a really hard time grasping the concepts in this subject. Thanks alot!