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

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