Question

Write the vector u = (1, −2, 5) ∈ R 3 as a linear combination of the vectors u1= (1, 1, 1), u2=

(1, 2, 3), u3= (2, −1, 1)

Answer 1

Let us determine the coefficients $c_1, c_2, c_3$ such as $u=c_1u_1 + c_2u_2 + c_3u_3.$ We may rewrite this equality as a system of linear equations, where each equation concerns one coordinate.

$\begin{cases} 1 &= c_1\cdot1 +c_2\cdot1 + c_3\cdot 2\\ -2 &= c_1\cdot1 +c_2\cdot2 + c_3\cdot (-1)\\ 5 &= c_1\cdot1 +c_2\cdot3 + c_3\cdot1\\ \end{cases}$

We should solve this system. First we subtract the first equation from the second and the third and obtain

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

Next, we subtract the second equation multiplied by two from the third equation

$\begin{cases} 1 &= c_1 +c_2 + 2c_3\\ -3 &= c_2 -3c_3\\ 10 &= 5 c_3\\ \end{cases}$

Therefore, $c_3 = 2, \;\; c_2 = -3+3\cdot2 = 3, \;\; c_1 = 1 - 3-2\cdot2 = -6.$

Therefore, $u = -6u_1 + 3u_2 + 2u_3.$