Toronto Math Forum

MAT244--2019F => MAT244--Lectures & Home Assignments => Chapter 7 => Topic started by: aremorov on November 10, 2019, 09:17:34 PM

Title: Finding linear independence
Post by: aremorov 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...
Title: Re: Finding linear independence
Post by: Victor Ivrii on November 11, 2019, 06:58:04 AM
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.