[Math] Is the vector in the span

Question

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?

Answer 1

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);