ECEN601_hw7_pfister - ECEN 601 Assignment 7 Problems 1 Let...

ECEN 601: Assignment 7
Problems:
1. Let V be the subspace of R [ x ] of polynomials of degree at most 3. Equip V with the inner
product
( f | g ) =
integraldisplay
1
0
f ( t ) g ( t ) dt.
(a) Find the orthogonal complement of the subspace of constant polynomials.
(b) Apply the Gram-Schmidt process to the basis { 1 ,x,x
2
,x
3
} .
2. Let W be a finite-dimensional subspace of an inner product space V , and let E be the
orthogonal projection of V on W . Prove that ( Eα | β ) = ( α | Eβ ) for all α , β in V .
3. Let S be a subset of an inner product space V . Show that ( S
⊥
)
⊥
contains the subspace
spanned by S . When V is finite-dimensional, show that ( S
⊥
)
⊥
is the subspace spanned by S .
4. Let X and Y be vector spaces over the same set of scalars. Let LT [ X,Y ] denote the set of
all linear transformations from X to Y . Let L and M be operators from LT [ X,Y ]. Define
an addition operator between L and M as
( L + M )( x ) = L ( x ) + M ( x )
for all x ∈ X . Also define scalar multiplication by
( aL )( x ) = a ( L ( x ))