Linear maps

A linear map is a function , where V and W are both vector spaces. They must satisfy the following criteria:

• , for all 
• , for all  and 

Linear maps tend to preserve the properties of vector spaces under addition and scalar multiplication. A linear map is called a homomorphism of vector spaces; however, if the homomorphism is invertible (where the inverse is a homomorphism), then we call the mapping an isomorphism. 

When V and W are isomorphic, we denote this as , and they both have the same algebraic structure.

If V and W are vector spaces in , and , then it is called a natural isomorphism. We write this as follows:

Here,  and  are the bases of V and W. Using the preceding equation, we can see that , which tells us that  is an isomorphism. 

Let's take the same vector spaces V and W as before, with bases  and  respectively. We know that  is a linear map, and the matrix T that has entries A_ij, where  and  can be defined as follows:

.

From our knowledge of matrices, we should know that the j^th column of A contains Tv_j in the basis of W.

Thus,  produces a linear map , which we write as .