the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups. An important example is the kernel of a linear map for some fixed matrix as above. The kernel of this map is the subspace of vectors such that which is precisely the set of solutions to the system of homogeneous linear equations belonging to This concept also extends to linear differential equations where the coefficients are functions in too. In the corresponding map the derivatives of the function appear linearly (as opposed to for example). Since differentiation is a linear procedure (that is, and for a constant ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation form a vector space (over or ). Direct product and direct sum The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The direct product of a family of vector spaces consists of the set of all tuples which specify for each index in some index set an element of Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum (also called coproduct and denoted ), where only tuples with finitely many nonzero vectors are allowed. If the index set is finite, the two constructions agree, but in general they are different. Tensor product The tensor product or simply of two vector spaces and is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map from the Cartesian product is called bilinear if is linear in both variables and That is to say, for fixed the map is linear in the sense above and likewise for fixed The tensor product is a particular vector space that is a universal recipient of bilinear maps as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors subject to the rules These rules ensure that the map from the to that maps a tuple to is bilinear. The universality states that given any vector space and any bilinear map there exists a unique map shown in the diagram with a dotted arrow, whose composition with equals This is called the universal property