Question

10. Determine whether or not, vectors ui(1,-2, 0, 3), u2 = (2, 3,0,-1), u3 = (3,9,-4,-2) e R is a linear combination of the (2,-1,2,1) 2

Answer 1

If (2, -1, 2, 1) is a linear combination of the three given vectors, then there should exist $c_1,c_2,c_3$ such that

$(2,-1,2,1)=c_1(1,-2,0,3)+c_2(2,3,0,-1)+c_3(3,9,-4,-2)$

or equivalently, there should exist a solution to the system

$\begin{cases}c_1+2c_2+3c_3=2\\-2c_1+3c_2+9c_3=-1\\-4c_3=2\\3c_1-c_2-2c_3=1\end{cases}$

Right away we get $c_3=-\dfrac12$, so the system reduces to

$\begin{cases}c_1+2c_2=\dfrac72\\\\-2c_1+3c_2=\dfrac72\\\\3c_1-c_2=0\end{cases}$

Notice that the first equation is the sum of the latter two. The third equation gives us

$3c_1-c_2=0\implies 3c_1=c_2$

so that in the second equation,

$-2c_1+3c_2=\dfrac72\implies7c_1=\dfrac72\implies c_1=\dfrac12$

which in turn gives

$3c_1=c_2\implies c_2=\dfrac32$

and hence the (2, -1, 2, 1) is a linear combination of the given vectors, with

$\boxed{(2,-1,2,1)=\dfrac12(1,-2,0,3)+\dfrac32(2,3,0,-1)-\dfrac12(3,9,-4,-2)}$