[Solved] MA502 Homework 5

1. Let X = C([0,1]) denote the space of continuous functions defined in the unit interval. Prove that the map.
2. Consider a basis of R³ 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 : Rⁿ² 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 = R₁ + R2 with R₁,R₂ Rⁿ and , R, then

D(A) = D(A₁) + D(A₂)

where A_i is the matrix constructed by taking A and replacing row R with tow R_i. This property is denoted as the determinant is a multilinear transformation row by row.

4. Prove that the determinant map D : Rⁿ² R defined above is alternating, i.e. if rows R_i and R_j in a matrix

R₁ R₁

R_i, ! R_j, !

A = are exchanged to obtain a new matrix A =

R_j R_i

R_n R_n

then D(A) = D(A).

5. Prove that for 22 matrices the determinant is the only map D : R⁴ 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.