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**preview**shows page 1. to view the full**4 pages of the document.**Tuesday11/01, lecture notes byY. Burda

1Fields

Todo linear algebra weonly need to do arithmetic operations to numbers. So

instead of numbers wecan work with anything wecan add, multiply,subtract

and divide (with the usual properties of the operations being assumed).

Example: the inverse of amatrix 1 2

3 4 can’t be−√2 1

3

2

1

2because ﬁnding

inverse matrix involves only arithmetic operations, the entries of the matrix

westarted with arerational and thus the answer should beamatrix with

rational entries only.

The relevantdeﬁnition here is that of aﬁeld:

Aset Kwith operation +and ·is called aﬁeld if:

Sum of twonumbers is anumber

x+y∈Kfor all x, y∈K

Order doesn’t matter for addition

x+y=y+xfor all x, y∈K

Grouping doesn’t matter for addition

(x+y)+z=x+(y+z) for all x, y,z∈K

Zero is anumber

there exists element0∈Ksuchthat x+0=xfor all x∈K

One can subtract numbers

for anyx∈Kthere exists −x∈Ksuchthat x+(−x)=0

Product of numbers is anumber

xy ∈Kfor all x, y∈K

Order doesn’t matter for multiplication

xy =yxfor all x, y∈K

Grouping doesn’t matter for products

(xy)z=x(yz)for all x, y,z∈K

One is in the ﬁeld

There exists 1∈Ksuchthat 1x=xfor all x∈K

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