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.

The given vector is $(6,1,-6,2)$ The vector set is $\{(1,-1,-1,0), (-1,0,1,1), (1,1,-1,1)\}$.

How does one prove that the vector is in the span of the set?

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

The vector $(6,1,-6,2)$ is in the span of $(1,-1,-1,0)$, $(-1,0,1,1)$, and $(1,1,-1,1)$ if and only if there exist real numbers $\alpha,\beta,\gamma$ such that $$(6,1,-6,2) = \alpha(1,-1,-1,0) + \beta(-1,0,1,1) +\gamma(1,1,-1,1).$$

This becomes a system of four equations in three unknowns, by looking at each coordinate in turn: $$\begin{array}{rcrcrcl} \alpha & - & \beta & + & \gamma & = & 6\\ -\alpha & & &+&\gamma &=& 1\\ -\alpha &+&\beta &-&\gamma &=&-6\\ &&\beta &+ &\gamma&=&2. \end{array}$$

If the system has a solution, this solution witnesses the fact that $(6,1,-6,2)$ lies in the span of the other three vectors. If the system has no solutions, then this shows the vector does not lie in the span of the other three.

P.S. Curly brackets, { and }, are usually used to denote sets, not vectors.

share|improve this answer
    
thank you Arturo, and thank you more for {} notation. –  Maysam Jul 2 '11 at 6:29
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.