# Is the vector in the span?

I have to show that $a_4$ is in the span of the other 3 vectors $a_1,a_2,a_3$

\begin{bmatrix} 2 &1 & 3\\ 3 & -1 & 4 \\ 1 & 2 & -1 \\ 4 & 4 & 3 \\ 0 & 3 & 5 \\ & \end{bmatrix} My Maple code: with (LinearAlgebra);

A := <<2, 1, 3>,<3, -1, 4>,<1, 2, -1>,<4, 4, 3>,<0, 3, 5>>; Then i use ReducedRowEchelon to get the linear combination which is M=:{{1,0,0},{0,1,0},{0,0,1},{0,0,0}, {0,0,0} }.

How do i prove that $a_4$ which is {-3,-6,5,2,7} is in the span of the other 3 vectors?

-
Oh, dear! Please use LaTeX here to write mathematics, otherwise things like your question are hard to read. Refer to the FAQ section for direction – DonAntonio Dec 17 '12 at 3:33
To do this, in general, I recommend checking out this previous answer. – process91 Dec 17 '12 at 3:34
So that meaning aa1+ba2+c*a3=a4 a(2,3,1,4,0)+b(1,-1,2,4,3)+c(3,4,-1,3,5)=a4 – user53368 Dec 17 '12 at 3:38
We already have a problem here, which may be a typo (a serious one) or simply a mistake in the question: all your vector are 3-dimensional...except $\,a_4\,$, which is 5-dimensiona (in spite of having written it with curly parentheses intead of <>, as the other ones). This can't be as $\,a_4\,$ doesn't even live in the same universe as the other vectors. – DonAntonio Dec 17 '12 at 3:39
Well I just copy pasted from Maple for that Vector. Gonna read up on Latex and try to do it in that format ASAP – user53368 Dec 17 '12 at 3:41

I am making an example here for you in Maple, so you can do yours as well. I think it inspires you well. Assume you have 3 vector as: $$u=\langle 2,1,-3,1 \rangle, v=\langle -1,3,5,0 \rangle,w=\langle 2,-1,1,-3 \rangle$$ and want to find scalars in which we can write the vector $$b=\langle -16,17,37,3 \rangle$$ as their combinations. You can do by:

[>with(linalg):

[> u:=matrix(4,1,[2,1,-3,1]):

[> v:=matrix(4,1,[-1,3,5,0]):

[> w:=matrix(4,1,[2,-1,1,-3]):

[> b:=matrix(4,1,[-16,17,37,3]):

[> A:=augment(u,v,w):

[> linsolve(A,b);

-
@GerryMyerson: Yes. Thanks for noting me that. – Babak S. Dec 17 '12 at 5:34
Nice...so you know Gap, and Maple, and ... ???? !!! – amWhy Apr 13 '13 at 0:35
@amWhy: Again, it is Maple :-) – Babak S. Apr 13 '13 at 4:56