Insanely Powerful You Need To Determinants
The determinant of a matrix A is denoted det(A), det A, or A.
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from
R
n
n
{\displaystyle \mathbf {R} ^{n\times n}}
to
R
{\displaystyle \mathbf {R} }
. The same idea is also used in the theory of differential equations: given functions
f
(
x
)
,
,
f
n
(
x
)
{\displaystyle f_{1}(x),\dots ,f_{n}(x)}
(supposed to be
n
1
{\displaystyle n-1}
times differentiable), the Wronskian is defined to be
It is non-zero (for some
x
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{\displaystyle x}
) in a specified interval if and only if the given functions and all their derivatives up to order
n
1
{\displaystyle n-1}
are linearly independent. .