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Fields, Vector Spaces and Vector Subspaces

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Martin, Burda

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Tuesday11/01, lecture notes byY. Burda
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 be2 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+yKfor all x, yK
Order doesn’t matter for addition
x+y=y+xfor all x, yK
Grouping doesn’t matter for addition
(x+y)+z=x+(y+z) for all x, y,zK
Zero is anumber
there exists element0Ksuchthat x+0=xfor all xK
One can subtract numbers
for anyxKthere exists xKsuchthat x+(x)=0
Product of numbers is anumber
xy Kfor all x, yK
Order doesn’t matter for multiplication
xy =yxfor all x, yK
Grouping doesn’t matter for products
(xy)z=x(yz)for all x, y,zK
One is in the field
There exists 1Ksuchthat 1x=xfor all xK
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