1) Determine if the following formulas define inner products on the given V by c

1)
Determine
if the following formulas define inner products on the given V by checking
whether or not the axioms in the definition of an inner product are satisfied.a)Let V = C [0 1]. Define
=f ( 1 ) g ( 1 ) .( Hint
:h (
x) = sin(x )is
continuous on [ 0 1 ] .b)
LetV = C [ 0 1 ] . define< f g > =( H
int :think of piecewise function .c)
Let
V =. D define< u v >by< u v > =+. d)LetAbe
anm x n
of rankn.V=.
define < u v> by< u v > = (Au ). ( Av ) where the
right hand side is the standard dot product in.( this is kind of a weirdexample because the calculation of the inner
product onis
being done on. we
did this in class form=n and Ainvertible.
But if Ais not square almost
exactly the same check goes through if and only if the matrixAhas
a rankn .