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Messages - aremorov

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Chapter 7 / Finding linear independence
« on: November 10, 2019, 09:17:34 PM »
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...

No, this is still possible to do if the determinant is 0.

For example:

$x_1' = x_1 + x_2$ (*)
$x_2' = x_1 + x_2$

has determinant 0 for the coefficients, however if we isolate for $x_1$ we get:
$x_1'' - 2x' = 0$ which has solution:
$x_1 = C_1 + C_2 e^{2t}$ and putting this into equation (*) gives us $x_2 = C_2 e^{2t} -C_1$ for arbitrary constants $C_1, C_2$.

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