Last Updated：2025/12/30
Sentence
Given
the
basis
of
some
vector
space
V,
how
to
find
its
dual
basis,
i.e.,
the
basis
of
the
dual
space
V^*?
Fill
the
columns
of
a
square
matrix
M
with
the
basis
vectors
of
V.
Find
the
inverse
matrix
M⁻¹
of
M.
Then
the
rows
of
M⁻¹
are
the
(co)vectors
of
that
dual
basis.
Since
(M⁻¹)⁻¹=M,
then
(V^*)^*=V.
Quizzes for review
Given the basis of some vector space V, how to find its dual basis, i.e., the basis of the dual space V*? Fill the columns of a square matrix M with the basis vectors of V. Find the inverse matrix M⁻¹ of M. Then the rows of M⁻¹ are the (co)vectors of that dual basis. Since (M⁻¹)⁻¹=M, then (V)^=V.
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