Formal Definitions and Algebraic Properties
The exterior algebra Λ(V) over a vector space V over a field K is defined as the quotient algebra of the tensor algebra by the two-sided ideal I generated by all elements of the form x ⊗ x such that x ∈ V. Symbolically,
The exterior product ∧ of two elements of Λ(V) is defined by
Read more about this topic: Exterior Algebra
Famous quotes containing the words formal, definitions, algebraic and/or properties:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“What I do not like about our definitions of genius is that there is in them nothing of the day of judgment, nothing of resounding through eternity and nothing of the footsteps of the Almighty.”
—G.C. (Georg Christoph)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)