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Leokris [45]
3 years ago
13

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
Mathematics
1 answer:
vodka [1.7K]3 years ago
5 0

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)}

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