Question

How would I prove a vector is in the span of a set of vectors?

Answer 1

if you can show that some linear combination of the vectors in the set make up the one you are asking about

Answer 2

So if I put them in a matrix and can get pivot positions in every row it would be in the span, right?

Answer 3

"pivot positions" ?
I don't know that term

Answer 4

Ah, well it pretty much means there is a solution to the given matrix. If there are pivots in every row then each vector can be multiplied by some number to get the answer. So I guess that's pretty much what you said, thanks!

Answer 5

say you have some set of vectors \( S\) such that\[\vec v_1=\langle x_1,y_1,z_1\rangle\]\[\vec v_2=\langle x_2,y_2,z_2\rangle\]\[\vec v_3=\langle x_3,y_3,z_3\rangle\]the \[\vec u\in S\iff\vec u=c_1\vec v_1+c_2\vec v_2+c_3\vec v_3\]for some\[c_1,c_2,c_3\in\mathbb R\]so if\[A\vec x=\vec u\]where \[\vec x=\langle c_1,c_2,c_3\rangle\]has some solution then \(\vec u\) is in the set.

Answer 6

...and \(A\) is the matrix formed by the vectors
so yeah, if that has a solution then \(\vec u\) is in \(S\)

Answer 7

*the span of \(S\)

Answer 8

Awesome, thanks so much. I'm just starting out linear algebra so the whole spanning concept is still new and shaky.

Answer 9

I like this site for all your supplementary needs
http://tutorial.math.lamar.edu/Classes/LinAlg/LinAlg.aspx

Answer 10

in particular you may want to look at this section
http://tutorial.math.lamar.edu/Classes/LinAlg/Span.aspx

Answer 11

wow thats awesome, thanks

Answer 12

welcome :)