Consider an algebra where the vector space is ℝ3 and
the multiplication of vectors is the conventional cross product you
learned as a beginning physics student. Find the structure
constants of this algebra.
Linear Algebra
we know that x ∈ R^n is a nonzero vector and C is a real
number.
find all values of C such that ( In − Cxx^T ) is nonsingular and
find its inverse
knowing that its inverse is of the same form
Course: Differential Geometry (Vector Calculus & Linear
Algebra)
Provide all proofs
(a) Find the formula for the distance from p to the line
y=mx
(b) prove that the set U={(x,y): y<mx} is an open
set
suppose that T : V → V is a linear map on a finite-dimensional
vector space V such that dim range T = dim range T2. Show that V =
range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range
T = {0}, and apply the fundamental theorem of linear maps.)
Differential Geometry (Mixed Use of Vector Calculus & Linear
Algebra)
1A. Prove that if p=(x,y) is in the set where y<x and if
r=distance from p to the line y=x then the ball about p of radius r
does not intersect with the line y=x.
1B. Prove that the set where y<c is an open set.
a) Using relevant algebra and a hypothetical example, explain
what the statement “the delta of a call option is 0.85” implies for
a bank that wants to hedge a position in the option.
b) Using relevant algebra, explain what the risks for option
writers facing a large position gamma while their portfolio is
delta hedged?
c) A hedge fund owns a portfolio of options on the US dollar–euro
exchange rate. The delta of the portfolio is 65. The current
exchange...