Grassmann 0011 0010 1010 1101 0001 0100 1011 Melissa Ray 1 Math 530 23 Dr. Chen 45 Agenda • Introduce Hermann Gunther Grassmann 0011 0010• Describe 1010 1101 0001 Grassmann’s 0100 1011 contributions to Linear Algebra 1 3 • The Impact on Linear Algebra today! 2 45 Hermann Gunther Grassmann • Born: April 15th, 1809 Stettin, Prussia 0011 0010 1010 1101 0001 0100 1011 • Father, Justus Gunther Grassmann a teacher1 of Mathematics and physics 3 • Mother, Johanne Grassmann2 45 As A Young Man… • Studied to be a minister with emphasis on language • Attempted to follow 0011 0010Maternal 1010 1101 Grandfather’s 0001 0100 1011 path of ministry 1 • Gained education in – Theology 3 – Literature 2 – Languages 45 The Change • Returned to Mathematics 0011 0010• Could 1010 1101 only 0001 teach 0100 1011 elementary age 1 • Seems desire was to make his parents 3 proud. 2 45 Grassmann’s Contributions • Publishes many works to be disregarded • In 1861, he proves arithmetic operations: – Commutativity 0011 0010 1010 1101 0001 0100 1011 – Associativity 1 – Distributivity. 3 • Most Importantly, develops the concept of2 Vector Space 45 Unfortunately…. • No language to explain vector space • Peono would later publish work expanding on Grassmann’s concept. 0011 0010 1010 1101 0001 0100 1011 • Was a loner – would accassionally 1 collaborate with Mobius, Hankel, and 3 Clebsch. 2 45 Vectors • Proves Vector Addition and Scalar vector Multiplication • Congruence not proven until after his death 0011 0010 1010 1101 0001 0100 1011 • More dimensions brought concepts 1named “inner product” and “outer product” 23 45 Inner Product • Represented as a “dot” • Determines the orthogonal projection through multiplication. 0011 0010 1010 1101 0001 0100 1011 • Inner product is not associative but 1is distributive, commutative and scalor 3 commutative. 2 45 Outer Product • Designates the dimensions of the geometric space and defines subspaces • Are associative, distributive in addition, and 0011 0010 1010 1101 0001 0100 1011 has commutative scalar multiplication1 • However, can be anti-communitive and anti 3 symmetric for coplaner vectors. 2 45 Where Grassmann left off: • William Kingdon Clifford expanded. • Peono attempted to cast light on the vector space concept. 0011 0010 1010 1101 0001 0100 1011 • Grassmann research only focused on1 the abstract properties, not the impact on 3 geometry 2 • Clifford would attribute most of his work to Grassmann. 45 Impact of Linear Algebra • Grassmann created the concept, the 20th century used it. 0011 0010 1010 1101 0001 0100 1011 • Geometry and Vector Space have a 1 common Ground 3 2 45 Conclusion 0011 0010• Did 1010 Grassmann 1101 0001 0100 1011live up to his Father’s expectations or surpass them? 123 45 Works Cited • Bookrags.com, "World of Mathematics on Hermann Gunther Grassmann". Web 6 October 2012. • Fearnley-Sander, Desmond., "Hermann Grassmann and the Creation of Linear Algebra". maa.org. Web 1 October 2012. • Hestenes, David., "Grassmann's Vision". Geocalcl.clas.asu.edu., 1996 0011 0010(Gert 1010 Schubring 1101 0001, ed.), 0100 Kluwer 1011 Academic Publishers, Dordrect/Boston 191-201. Web 1 October 2012. • aix1.uottawa.ca/~jkhoury/matrices.html, Web 16 November1 2012. • Schulz, William C., "Theory and application of Grassmann Algebra". Transgalactic Publishing Company. Flagstaff, Vienna, 31 August 3 2011 . Web 1 October 2012. 2 • Zaharia, Marius Dorian. "Computer Graphics from a Geometric Algebra Perspective". Intelligent Autonomous Systems. Netherlands. Web 1 October 2012. 45.