Answer to Question #296000 in Linear Algebra for Mayank Chakre

Question

Write the vector (1, −2, 5) as a linear combination of the vectors (1, 1, 1),(1, 2, 3)

and (2, −1, 1)

Answer 1

Let us write the vector $(1, −2, 5)$ as a linear combination of the vectors $(1, 1, 1),(1, 2, 3)$

and $(2, −1, 1).$

Let

 $(1, −2, 5)=a(1, 1, 1)+b(1, 2, 3)+c(2, −1, 1)=(a+b+2c,a+2b-c,a+3b+c).$

It follows that we get the system

$\begin{cases} a+b+2c=1\\ a+2b-c=-2\\ a+3b+c=5 \end{cases}$

which is equivalent after subtracting from the second row the first row, and from the third row the second row to the system

$\begin{cases} a+b+2c=1\\ b-3c=-3\\ b+2c=7 \end{cases}$

and hence the last system is equivalent after subtracting from the third row the second row to the system

$\begin{cases} a+b+2c=1\\ b-3c=-3\\ 5c=10 \end{cases}$

It follows that

 $c=2,\\ b=3c-3=6-3=3,\\ a=1-b-2c=1-3-4=-6.$

We conclude that $(1, −2, 5)=-6(1, 1, 1)+3(1, 2, 3)+2(2, −1, 1).$