Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose I have $\sum_{i = 1}^{n} v_{i} = \sum_{i = 1}^{n} w_{i}$ where $\{v_{i}\}_{i = 1}^{n}$ and $\{w_{j}\}_{j = 1}^{n}$ are both sets of linearly independent vectors in $\mathbb{R}^{d}$ ($d$ possibly $\neq n$). Is it true that we must have $\{v_{i}\}_{i = 1}^{n} = \{w_{j}\}_{j = 1}^{n}$?

share|improve this question
2  
$(0,1)+(1,0) = (1/2, 0)+(1/2, 1)$. –  David Mitra Feb 21 '12 at 0:37
add comment

2 Answers

up vote 2 down vote accepted

No; for example, let $d=n=2$, and let $$v_1=(0,1),\;v_2=(2,1)\quad\text{ and }\quad w_1=(1,2),\; w_2=(1,0).$$ Then $\{v_1,v_2\}\neq\{w_1,w_2\}$ but $$v_1+v_2=(0,1)+(2,1)=(2,2)=(1,2)+(1,0)=w_1+w_2.$$

share|improve this answer
add comment

No it is not. Take $n=2$. Take $\left\{\begin{bmatrix}1\\1\end{bmatrix},\begin{bmatrix}1\\-1\end{bmatrix}\right\}$ and $\left\{\begin{bmatrix}2\\1\end{bmatrix},\begin{bmatrix}0\\-1\end{bmatrix}\right\}$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.