Question

Write the vector v=( 1, -2, 5) as a linear combination of the vector e1=( 1, 1, 1), e2= ( 1, 2, 3) and e3=( 2, -1, 1)

Answer 1

Let us assume $v=c_1e_1 + c_2e_2 + c_3e_3.$ We may write a linear system for every coordinate:

$\begin{cases} 1 = c_1+c_2+2c_3, \\ -2 = c_1+2c_2-c_3, \\ 5=c_1+3c_2+c_3. \end{cases}$

We should solve this linear system and get $c_1 = -6, \; c_2 = 3, \; c_3 =2.$

Therefore, $v = -6e_1+3e_2+2e_3.$