1. ## Linear Independence

Let $\displaystyle [v_1, v_2, v_3]$ be a linearly independent subset of a vector space over $\displaystyle C$. Determine
all scalars $\displaystyle \delta$ with which $\displaystyle [v_1-\delta{v_2}, \delta{v_1} + v2, v_1 + v_2 + v_3]$ is linearly independent.

Attempt so far

If $\displaystyle [v_1, v_2, v_3]$ is linearly independent, then for $\displaystyle a_i\in{C}$
$\displaystyle a_1v_1+a_2v_2+a_3v_3\neq{0}$
And consider
$\displaystyle a_1v_1+a_2v_2+a_3v_3+v_1(\delta{a_2}+a_3)+v_2(-\delta{a_1}+a_3)\neq{0}$
How do I prove linear independence from here?

2. Originally Posted by I-Think
Let $\displaystyle [v_1, v_2, v_3]$ be a linearly independent subset of a vector space over $\displaystyle C$. Determine
all scalars $\displaystyle \delta$ with which $\displaystyle [v_1-\delta{v_2}, \delta{v_1} + v2, v_1 + v_2 + v_3]$ is linearly independent.

Attempt so far

If $\displaystyle [v_1, v_2, v_3]$ is linearly independent, then for $\displaystyle a_i\in{C}$
$\displaystyle a_1v_1+a_2v_2+a_3v_3\neq{0}$
And consider
$\displaystyle a_1v_1+a_2v_2+a_3v_3+v_1(\delta{a_2}+a_3)+v_2(-\delta{a_1}+a_3)\neq{0}$
How do I prove linear independence from here?
Hint: consider the determinant of the matrix

$\displaystyle \begin{vmatrix}1 & \delta & 1 \\ -\delta & 1 & 1\\ 0 & 0 & 1 \end{vmatrix}=1+\delta^2$

Why does this help?