# Why is a vector a rank-1 tensor?

I learned that a tensor is a linear map from some number of vectors to the real numbers.

For example, $$t(\vec{u},\vec{v})=\vec{u}\cdot\vec{v}$$ is a rank-2 tensor as it takes 2 vectors as arguments and combines them into a real number.

I then read that a rank-1 tensor is a vector. How is that possible? If the rank-1 tensor is a vector, it will mean $$t({\vec u})=\vec{u}$$, but $$\vec{u}$$ is not a real number.

• "If the rank-1 tensor is a vector, it will mean $t({\vec u})=\vec{u}$" How so? – Acccumulation Sep 20 at 19:54

An $$(r,s)$$-tensor maps $$r$$ dual vectors and $$s$$ vectors to some field (e.g. $$\mathbb{R}$$). The rank of a tensor is defined as $$r+s$$. One can interpret a vector $$v$$ as $$(1,0)$$-tensor, mapping a dual vector $$w$$ to the number $$w(v)$$.