Finding linear independence

[aremorov]

This is related to section 7.3 (specifically question 13 in the module).

How would we show whether the vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$ are linearly independent, where:

$\vec{v_1} = \begin{bmatrix}e^t \\ e^{3t}\end{bmatrix}$,
$\vec{v_2} = \begin{bmatrix}e^{4t} \\ e^{5t}\end{bmatrix}$
$\vec{v_3} = \begin{bmatrix}e^{2t} \\ e^{7t}\end{bmatrix}$

Taking the determinant $|\vec{v_1} \quad \vec{v_2} \quad \vec{v_3}|$ doesn't make sense...

[Victor Ivrii]

Those are not just vectors, but vector-valued functions and you need to check that for constant coefficients  their linear combination is identically $0$ if and only if these coefficients are $0$.
Try first to look at components of this vector-function.