- #1

- 40

- 14

1. Homework Statement

1. Homework Statement

Let

**x**and

**y**be vectors in ℝ

^{n}. Is it possible that

**x**is a scalar multiple of

**y**(i.e., there exists a scalar c such that

**x**= c

**y**), but

**y**is

*not*a scalar multiple of

**x**?

## Homework Equations

Basically restating the problem in an equation here, from what I see no pure equation other than this is needed:

Let b, c be scalars in ℝ and

**x**,

**y**be vectors in ℝ

^{n}. Let the scalar c be defined such that

**x**= c

**y**. Is there always a b such that

**y**= b

**x**?

## The Attempt at a Solution

There is not always a scalar b for the given vectors

**x**,

**y**and given scalar c to make the equations above true. Take the case c = 0,

**x**= (0,0), y ≠ (0,0). Then (0,0) = 0

**y**. But there does not exist a scalar b such that b

**x**= b⋅(0,0) =

**y**≠ (0, 0)

*by the fact that (0,0)⋅b = (0,0) for all b.*I believe I have interpreted the question correctly, and it seems that the first part of my "proof" (may not be completely formal or rigorous - I'll be open to any suggestions to improve it) is valid by the proof here. But I cannot figure out if the underlined statement is simply a postulate of linear algebra, if there is a way that I should prove it, or if it is even correct. Should the last statement of the proof be changed or is it even valid?

Thanks to anyone who takes the time to help.

EDIT: Clarity in last paragraph.