Question

MATRECIES: Suppose ={u,v,w}⊆R3 is linearly independent. Determine if the set 4 ={u, v, v−w, u+v+w} is linearly independent or linearly dependent.

Answer 1

The set is linearly dependent.

To explicitly prove this, we need to show there is at least one choice of constants $c_1,c_2,c_3,c_3\in\Bbb R$ such that

$c_1 u + c_2 v + c_3(v-w) + c_4(u+v+w) = 0$

or equivalently,

$(c_1 + c_4) u + (c_2 + c_3 + c_4) v + (-c_3 + c_4) w = 0$

which is the same as solving the system of equations

$\begin{cases} c_1 + c_4 = 0 \\ c_2 + c_3 + c_4 = 0 \\ -c_3 + c_4 = 0 \end{cases}$

From the first and last equations, we have $c_1=-c_4$ and $c_3=c_4$. Substituting these into the second equation leaves us with $c_2-2c_1=0$, and so the overall solution set is

$c_1 = \dfrac12 c_2 = -c_3 = -c_4$

for which there are infinitely many not-all-zero solutions.