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.
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 .