Question

Letv1 = (0,3,6,0),v2 = (0,2,4,6), and v3 = (1,−1,−2,1). Express (4,−1,−2,−11) as a

linear combination of v1,v2, and v3

Answer 1

To find a linear combination we should solve a system

$\begin{bmatrix} 0 & 0 & 1 \\ 3 & 2 & -1 \\ 6 & 4 & -2 \\ 0 & 6 & 1 \end{bmatrix} \cdot \begin{bmatrix} x1 \\ x2 \\ x3 \end{bmatrix} = \begin{bmatrix} 4 \\ -1 \\ -2 \\ -11 \end{bmatrix}$

That is equivalent to the system of four equations:

$\begin{cases} x3 = 4 \\ 3x1+2x2-x3 = -1 \\ 6x1 +4x2 -2x3 = -2 \\ 6x2+x3 = -11 \end{cases}$

From the first equation get x3 = 4. Substitute this value into the fourth equation obtain 6 x2 = -11 -4, or x2 = -5/2. And from the second equation 3 x1 = -1 +5 +4, so x1 = 8/3. Substituting these values into the third equation, we make sure that it is also fulfilled.

So (4,−1,−2,−11) = 8*v1 -5/2*v + 4*v3