## 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 **shift** of **hyperplane**.

**Exercise 3**. As we know, the equation

**Proof**. Let

**Conclusion**. If

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