There is an answer of the question "Why is the tensor product important when we already have direct and semidirect products?" which states that

they allow you to study certain non linear maps (bilinear maps) by transforming them first into linear ones, to which you can apply linear algebra;---Mariano Suárez-Álvarez♦

Could anyone give me some concrete (and basic as possible, without using category theory) examples to show this application?

Please don't just show that there is a linear map which is induced by the multilinear map. Please show that how can we use the **linearity** of the linear map to gain some information or properties of the multilinear map.