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Lecture

# Fields, Vector Spaces and Vector Subspaces

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University of Toronto St. George

Mathematics

MAT224H1

Martin, Burda

Winter

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Tuesday 1101, lecture notes by Y. Burda 1 Fields To do linear algebra we only need to do arithmetic operations to numbers. So instead of numbers we can work with anything we can add, multiply, subtract and divide (with the usual properties of the operations being assumed). 2 1 Example: the inverse of a matrix 3 4 cant be 3 1 because nding 2 2 inverse matrix involves only arithmetic operations, the entries of the matrix we started with are rational and thus the answer should be a matrix with rational entries only. The relevant denition here is that of a eld: A set K with operation + and is called a eld if: Sum of two numbers is a number x + y K for all x,y K Order doesnt matter for addition x + y = y + x for all x,y K Grouping doesnt matter for addition (x+y)+z=x+(y+z) for all x,y,z K Zero is a number there exists element 0 K such that x + 0 = x for all x K One can subtract numbers for any x K there exists x K such that x + (x) = 0 Product of numbers is a number xy K for all x,y K Order doesnt matter for multiplication xy = yx for all x,y K Grouping doesnt matter for products (xy)z = x(yz) for all x,y,z K One is in the eld There exists 1 K such that 1x = x for all x K 1 www.notesolution.com

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