MA502 - Homework 5.
1. Let X = C([0, 1]) denote the space of continuous functions defined in
the unit interval. Prove that the map T(g) = R 1
0
g(x)dx is in X∗
.
2. Consider a basis of R
3
composed of the vectors
(1, 0, −1), (1, 1, 1) and (2, 2, 0)
find its dual basis.
3. Prove that the determinant, interpreted as a transformation
D : R
n
2 → R with D(A) = determinant(A)
is linear in each of the rows. That is, if a row R of the matrix A is
given by R = αR1 + βR2 with R1, R2 ∈ R
n and α, β ∈ R, then
D(A) = αD(A1) + βD(A2)
where Ai
is the matrix constructed by taking A and replacing row R
with tow Ri
. This property is denoted as the determinant is a multilinear transformation row by row.
4. Prove that the determinant map D : R
n
2 → R defined above is alternating, i.e. if rows Ri and Rj
in a matrix
A =
R1
...
Ri
,
...
Rj
...
Rn
!
are exchanged to obtain a new matrix A˜ =
R1
...
Rj
,
...
Ri
...
Rn
!
then D(A) = −D(A˜).
5. Prove that for 2×2 matrices the determinant is the only map D : R
4 →
R that is both multilinear as a function of the 2 rows and alternating,
and that takes the value D(I) = 1 at the identity. The proof can be
1
done directly, using multilinearity and the alternating property. Just
write any row in the matrix as a sum of vectors in the canonical basis.
Note This result, a characterization of the determinant, holds in any
dimensions and can be used as an alternative (and equivalent) definition
of the determinant.
2
