PMATH 900: Special Algebraic Structures
Assignment 5; due Thursday, 05 April 2012
[1] Let O be the algebra of the octonions, and let be the 3-form on Im O dened by (a, b, c) = ab, c .
[a] Prove that
1
|(a, b, c)|2 + |[a, b, c]|2 = |a b c|2
4
for all a, b,

PMATH 900: Special Algebraic Structures
Assignment 4; due Friday, 23 March 2012
[1] Let A be a (possibly nonassociative) nite-dimensional algebra with identity 1 over the eld R of
real numbers. Suppose that A has an inner product , (a nondegenerate symmet

PMATH 900: Special Algebraic Structures
Assignment 2; due Friday, 10 February 2012
[1] Let V be an n-dimensional vector space over a commutative eld F . Let G be a symmetric bilinear
form on V . Recall that we proved there exists a basis B = cfw_e1 , . .

PMATH 900: Special Algebraic Structures
Assignment 3; due Monday, 05 March 2012
Unless noted otherwise, throughout this assignment, V is an n-dimensional vector space over R, and
g(, ) = , is a metric on V . That is, g is a nondegenerate symmetric bilinea

PMATH 900: Special Algebraic Structures
Assignment 1; due Friday, 20 January 2012
[1] Let F be a (possibly noncommutative) eld, and let V and W be right F -vector spaces. Let L(V, W )
be the set of all right F -linear maps from V to W . That is, if T L(V,