Tensors can be described as multilinear mappings from r copies of dual vector space (V^*) and s copies of vector space V. Multilinearity means linear in each variable. Thus you can take σ and δ from dual vector space while v can be taken from vector space V. Because it is multilinear in each argument, this can be written as
(σ + δ, v) = (σ, v) + (δ, v)
From beginning you have (2, 1) on the left side of the equation but because of the multilinearity you have two (1,1) tensors on the right side of the equation. Plz let me know if I am wrong.