I'm having some trouble internalizing the concept of span.(adsbygoogle = window.adsbygoogle || []).push({});

The question:

Ifu= [1,2,1];v= [-2,2,4]; andw= [-1,4,5], describe Span{u,v,w}.

The attempt at a solution:

I formed a matrix using column vectorsu,v, andwand row-reduced to RREF:

[itex]

\begin{bmatrix}

1 & -2 & -1 \\

2 & 2 & 4 \\

1 & 4 & 5

\end{bmatrix}

[/itex] ~ [itex]

\begin{bmatrix}

1 & 0 & 1 \\

0 & 1 & 1 \\

0 & 0 & 0

\end{bmatrix}[/itex]

1) Given the equation Ax=bwith solutions x_{1}, x_{2}, and x_{3}, wherebis the vector [b_{1}, b_{2}, b_{3}] and b_{3}must = 0 in order for the equation to be consistent, does that mean Span{u,v,w} must lie in the plane x_{3}= 0 in R^{3}?

2) Since I saw that b_{3}= 0 was the only condition for this system to be consistent, I assumed Span{u,v,w} had to be a plane. Is there any way for me to see this parametrically by seeing that there are two free variables? I don't know if that even makes sense to ask...

So far in class we have been able to describe solution sets to matrix equations parametrically, so if the parametric description for the solution set of some equation was in the form x_{1}v_{1}+ x_{2}v_{2}, I could see that x_{1}and x_{2}are the free variables (and x_{3}is the basic variable). So in this case I could see geometrically that the solution set is a plane. Is there an analogous way to interpret the span?

By the way, we have not talked about vector spaces yet so that probably wouldn't help me in an explanation... thanks so much in advance!

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# Parametric description of span

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