Grassmann 0011 0010 1010 1101 0001 0100 1011 Melissa Ray 1 Math 530 23 Dr. Chen 45 Agenda 0011 0010 1010 1101 0001 0100 1011 • Introduce Hermann Gunther Grassmann

• Describe Grassmann’s contributions to 1 3 • The Impact on Linear Algebra today! 2 45 Hermann Gunther Grassmann

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• Born: April 15th, 1809 Stettin,

• Father, Justus Gunther Grassmann a teacher1 of and 3 • Mother, Johanne Grassmann2 45 As A Young Man…

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• Studied to be a minister with emphasis on language • Attempted to follow Maternal Grandfather’s path of ministry 1 • Gained education in – Theology 3 – Literature 2 – Languages 45 The Change 0011 0010 1010 1101 0001 0100 1011 • Returned to Mathematics

• Could only teach elementary age 1 • Seems desire was to make his parents 3 proud. 2 45 Grassmann’s Contributions 0011 0010 1010 1101 0001 0100 1011 • Publishes many works to be disregarded • In 1861, he proves arithmetic operations: – Commutativity – Associativity 1 – Distributivity. 3 • Most Importantly, develops the concept of2 45 Unfortunately…. 0011 0010 1010 1101 0001 0100 1011 • No language to explain vector space • Peono would later publish work expanding on Grassmann’s concept. • Was a loner – would accassionally 1 collaborate with Mobius, Hankel, and 3 Clebsch. 2 45 Vectors 0011 0010 1010 1101 0001 0100 1011 • Proves Vector Addition and Scalar vector Multiplication • Congruence not proven until after his death • More brought concepts 1named “inner product” and “outer product” 23 45 Inner Product 0011 0010 1010 1101 0001 0100 1011 • Represented as a “dot” • Determines the orthogonal projection through multiplication. • Inner product is not associative but 1is distributive, commutative and scalor 3 commutative. 2 45 Outer Product 0011 0010 1010 1101 0001 0100 1011 • Designates the dimensions of the geometric space and defines subspaces • Are associative, distributive in addition, and has commutative scalar multiplication1 • However, can be anti-communitive and anti 3 symmetric for coplaner vectors. 2 45 Where Grassmann left off: 0011 0010 1010 1101 0001 0100 1011 • expanded. • Peono attempted to cast light on the vector space concept. • Grassmann research only focused on1 the abstract properties, not the impact on 3 2 • Clifford would attribute most of his work to Grassmann. 45 Impact of Linear Algebra 0011 0010 1010 1101 0001 0100 1011 • Grassmann created the concept, the 20th century used it.

• Geometry and Vector Space have a 1 common Ground 3 2 45 Conclusion 0011 0010 1010 1101 0001 0100 1011

• Did Grassmann live up to his Father’s expectations or surpass them? 123 45 Works Cited 0011 0010 1010 1101 0001 0100 1011 • 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 (Gert Schubring, ed.), Kluwer 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 Perspective". Intelligent Autonomous Systems. Netherlands. Web 1 October 2012. 45