- #1

- 248

- 0

## Homework Statement

Let E

_{1}= (1, 0, ... ,0), E

_{2}= (0, 1, 0, ... ,0), ... , E

_{n}= (0, ... ,0, 1)

be the standard unit vectors of R

^{n}. Let x

_{1}... ,x

_{n}be numbers. Show that if

x

_{1}E

_{1}+...+x

_{n}E

_{n}=

**0**then x

_{i}=

**0**for all i.

## Homework Equations

## The Attempt at a Solution

Proof By contradiction

Assume to the contrary that x

_{1}E

_{1}+...+x

_{n}E

_{n}=

**0**then x

_{i}≠

**0**for some i. We also assume that x

_{1}...x

_{i-1}and x

_{i+1}...x

_{n}are zero. Rewriting the equation we get

x

_{1}E

_{1}+.x

_{p}E

_{p}+...+x

_{n}E

_{n}=

**0**where x

_{p}E

_{p}is a nonzero scalar. x

_{p}E

_{p}=-x

_{1}E

_{1}-...-x

_{p}E

_{p-1}-x

_{p}E

_{p+1}-..-x

_{n}E

_{n}. But this leads to a contradiction since we assumed earlier that x

_{1}...x

_{i-1}and x

_{i+1}...x

_{n}are zero. Thus x

_{1}E

_{1}+...+x

_{n}E

_{n}=

**0**x

_{i}=

**0**for all i.

Let me know where my proof begins to fall apart? And how do I go about it?