Geometry of linear equations: structure of image and null space
Definition 1. The subspace notion allows us to describe the algebraic structure of the set of solutions of The special case is called a homogeneous equation. Obviously, satisfies it but there may be other solutions. The equation is called an inhomogeneous equation. We address the questions of existence and uniqueness of its solutions.
Structure of the image of
Recall Basic observation 1: The image is the set of for which the inhomogeneous equation has solutions.
Exercise 1. is a linear subspace in
Proof. This follows from the first characterization of the matrix image. Here is a direct proof. Suppose Then there exist such that By linearity this gives Thus for we have found such that which means for any and is a subspace.
Structure of the null space of
Definition 2. The set of solutions of the homogeneous equation is called the null space of It is denoted
Exercise 2. The null space of is a linear subspace of
Proof. Suppose so that Then by linearity so for any and is a linear subspace.
Description of the set of solutions of
Intuition. In straight lines and planes that don't contain the origin can be obtained by shifting straight lines and planes that do (geometry should dominate the algebra at this point, see Figure 1). This is generalized in the next definition.
Definition 3. Let be any vector and let be a subspace. denotes a shift of by and it is obtained by adding to all elements of Some people call a hyperplane.
Exercise 3. As we know, the equation has solutions if and only if Let us fix and let be some solution of . Then the set of all solutions of this equation is (in detail: any other solution of that equation can be obtained by adding an element of the null space to : ). This is written as
Proof. Let and let be any solution of Then and for some We obtain which proves the inclusion Conversely, if with , then and we obtain
Conclusion. If , then is one-to-one and we have uniqueness of solutions of the inhomogeneous equation; otherwise, can serve as a measure of non-uniqueness.