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In algebra , a primordial element is a particular kind of a vector in a vector space .
Let
V
{\displaystyle V}
field
F
{\displaystyle \mathbb {F} }
(
e
i
)
i
∈
I
{\displaystyle \left(e_{i}\right)_{i\in I}}
I
{\displaystyle I}
basis of vectors for
V
.
{\displaystyle V.}
v
∈
V
{\displaystyle v\in V}
v
=
∑
i
∈
I
a
i
(
v
)
e
i
{\displaystyle v=\sum _{i\in I}a_{i}(v)e_{i}}
I
{\displaystyle I}
(
a
i
)
i
∈
I
{\displaystyle \left(a_{i}\right)_{i\in I}}
a
i
{\displaystyle a_{i}}
I
(
v
)
=
{
i
∈
I
:
a
i
(
v
)
≠
0
}
{\displaystyle I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}}
v
{\displaystyle v}
W
{\displaystyle W}
V
,
{\displaystyle V,}
p
∈
W
{\displaystyle p\in W}
primordial if it has both of the following two properties:[ 1]
I
(
p
)
{\displaystyle I(p)}
I
(
w
)
,
{\displaystyle I(w),}
0
≠
w
∈
W
,
{\displaystyle 0\neq w\in W,}
a
i
(
p
)
=
1
{\displaystyle a_{i}(p)=1}
i
.
{\displaystyle i.}
Linear equations Matrices
Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Spectrum Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Jordan normal form Weyr canonical form Rank Inverse , Pseudoinverse Adjugate , Transpose Dot product Symmetric matrix , Skew-symmetric matrix Orthogonal matrix , Unitary matrix Hermitian matrix , Antihermitian matrix Positive-(semi)definite Pfaffian Projection Spectral theorem Perron–Frobenius theorem Diagonal matrix , Triangular matrix , Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix , Hessian matrix Vandermonde matrix Stochastic matrix , Toeplitz matrix , Circulant matrix , Hankel matrix (0,1)-matrix List of matrices Matrix decompositions Relations and computations Vector spaces Structures Multilinear algebra Affine and projective Numerical linear algebra
