- #1

zeion

- 466

- 1

## Homework Statement

Let [tex] \mathbf {a} \in R^n [/tex] be a non zero vector, and define {[tex] S = \mathbf {x} \in R^n : \mathbf {a} \cdot \mathbf {x} = 0 [/tex] }. Prove that S interior [tex] = {\o} [/tex]

## Homework Equations

## The Attempt at a Solution

Intuitively I understand that if a is a vector in R^3, S would be a plane. And if I place an open ball on any point on the plane it would include points that are not on the plane. So S interior would be empty.

But I'm not sure how to prove this rigorously?

I'm supposed to somehow use the hint that a hyperplane in R^n will have 1 dimension less than R^n (like the plane in R^3 has 2 dimensions)

But I am confused about the dimension of the plane? All vectors on a plane in R^3 will have 1 (the same) component = 0? So does this mean that the dimension of the plane is 2?