I dont have much of a background in linear algebra and am trying to solve a problem for a vector analysis class.
I am trying to prove that a certain basis is linearly independent.
Lets say I have an orthonormal basis
$\hat{e_1}$, $\hat{e_2}$, $\hat{e_3}$
then change to another basis where the new basis vectors are combinations of the original (change of basis?)
How do i show that the new basis vectors are linearly independent?
Is it as simple as checking for a non zero determinant of the matrix where the rows are the new basis vectors?
Let say you have three new vectors $v_1,v_2,v_3$. Then they are linear combination of $\hat{e_1},\hat{e_2},\hat{e_3}$, ie $$ v_i = \sum\limits_{j=1}^3 a_{ij} \hat{e_j} $$ In other words, the coordinates of the $v_i$ with respect to the basis $\{\hat{e_1},\hat{e_2},\hat{e_3}\}$ form the rows of the matrix $$ A:=\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} $$ Hence $\det A =0 $ if and only if its rows are linearly dependent if and only if the $v_i$ are linearly dependent.