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.

Is there an example of a $3\times 3$ matrix $M = [v_{1}\, v_{2}\, v_{3}]$ (where $v_{i}$ are $3 \times 1$ vectors) such that $v_{1}, v_{2}, v_{3}$ are linearly dependent however if we have $av_{i} + bv_{j} = 0$ then $a, b = 0$ for each $(i, j)$ pair?

share|improve this question

3 Answers 3

In general, any three vectors that span a two dimensional space, but such that there is no vector a multiple of another.

For a subset of two such vectors will still span a two dimensional space, hence be linearly independent.

Both other answers fall in this category.

share|improve this answer

Yes, $v_1=[1,2,3]$, $v_2=[3,2,1]$ and $v_3=[4,4,4]$. Clearly the three vectors are dependent. However it is trivial to check that pairwise they are independent.

share|improve this answer

Yes: $$v_1 = (1,0,0)$$ $$v_2 = (1,1,0)$$ $$v_2 = (0,1,0)$$

share|improve this answer

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.