Let us look at the first class of multilinear maps; the bilinear maps. We want to study bilinear maps. The notion of "study" will have different meanings as we move forward (or backward) in the course. Let \(V,W,T\) be vector spaces and \(\varphi:V\times W\rightarrow T\) be a bilinear map. The feeling that "we are good at linear algebra" suggests us to ask the question : ...
In group theory, we mainly study maps that preserve the group structures; which goes by the name of group homomorphisms. In topology, we mainly study maps that preserve the topologies; which goes by the name of continuous functions. In theory of vector spaces, we mainly study maps that preserve the vector space structures; which goes by the name of linear maps. Apart from that, there are many interesting maps that comes up when dealing with vector spaces which are not really linear maps. The very first example that comes to mind is the determinant map ...
Let \(V\) be a vector space over \(\mathbb{R}\). Let \(\{e_1,\cdots,e_r\}\) be a basis of \(V\) over \(\mathbb{R}\) and \(\{e^1,\cdots,e^r\}\) be the dual basis of \(V\). We call \(e^i:V\rightarrow \mathbb{R}\) to be polynomials over \(V\) with values in \(\mathbb{R}\). A map \(p:V\rightarrow \mathbb{R}\) is said to be a polynomial map if \(p=\sum a_{t_1,\cdots,t_r}(e^1)^{t_1}\cdots(e^r)^{t_r}\). Let \(f:V\times V\times \cdots\times V\rightarrow \mathbb{R}\) be a symmetric multilinear mapping. We want to associate