Problem 1: Prove that
Problem 2: When does equality hold in the following theorem? ? Hint: Re-examine the proof of the theorem, the answer is not “when x and y are linearly dependent.”
Problem 3: Prove that . When does inequality hold?
Problem 4: Prove that ?
Problem 5: The quantity is called the distance between x and y. Prove and interpret geometrically the “triangle inequality” :
.
Problem 6: Let functions f and g be integrable on .
(a) Prove that . Hint: Consider separately the cases
for some
and
for all
.
(b) If equality holds, must for some
? What if f and g are continuous?
(c) Show that the following theorem is a special case of (a) above: , equality holds if and only if x and y are linearly dependent.
Problem 7: A linear transformation is norm preserving if
amd inner product preserving if
(a) Prove that T is norm preserving if and only if T is inner product preserving.
(b) Prove that such a linear transformation T is and
is of the same sort.
Problem 8:
If are non-zero, the angle between x and y, denoted
is defined as
, which makes sense by the following theorem :
The linear transformation T is angle preserving if T is 1-1, and for we have
(a) Prove that if T is norm preserving, then T is angle preserving.
(b) If there is a basis of
and numbers
such that
, prove that T is angle preserving if and only if all
are equal.
(c) What are all angle preserving ?
Problem 9: If , let
have the matrix
.
Show that T is angle preserving and if , then
Problem 10: If is a linear transformation, show that there is a number M such that
for
. Hint: Estimate
in terms of
and the entries in the matrix of T.
Problem 11: If and
, show that
and
. Note that
and
denote points in
.
Problem 12: Let denote the dual space of the vector space
. If
, define
by
. Define
by
. Show that T is a 1-1 linear transformation and conclude that every
is
for a unique
.
Problem 13: If , then x and y are called perpendicular (or orthogonal) if
. If x and y are perpendicular, prove that
.
Regards,
Nalin Pithwa