Linear subspace/Solution space for linear system/Introduction/Section

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Let be a field, and be a -vector space. A subset is called a linear subspace, if the following properties hold.

  1. .
  2. If , then also .
  3. If and , then also holds.

Addition and scalar multiplication can be restricted to such a linear subspace. Hence, the linear subspace is itself a vector space, see exercise. The simplest linear subspaces in a vector space are the null space and the whole vector space .

Let be a field, and let

be a homogeneous system of linear equations over . Then the set of all solutions to the system is a linear subspace of the standard space .


Therefore, we talk about the solution space of the linear system. In particular, the sum of two solutions of a system of linear equations is again a solution. The solution set of an inhomogeneous linear system is not a vector space. However, one can add, to a solution of an inhomogeneous system, a solution of the corresponding homogeneous system, and get a solution to the inhomogeneous system again.

We have a look at the homogeneous version of example, so we consider the homogeneous linear system

over . Due to fact, the solution set is a linear subspace of . We have described it explicitly in example as

which also shows that the solution set is a vector space. With this description, it is clear that is in bijection with , and this bijection respects the addition and also the scalar multiplication (the solution set of the inhomogeneous system is also in bijection with , but there is no reasonable addition nor scalar multiplication on ). However, this bijection depends heavily on the chosen "basic solutions“ and , which depends on the order of elimination. There are several equally good basic solutions for .

This example shows also the following: the solution space of a linear system over is "in natural way“, that means independent on any choice, a linear subspace of (where is the number of variables). For this solution space, there always exists a "linear bijection“ (an "isomorphism“) to some (), but for is no natural choice for such a bijection. This is one of the main reasons to work with abstract vector spaces, instead of just .