1 Types of Numbers
algebra:
algebra is doing stuff to things
- idea of a number changes – 500yago they didnt know about negs
natural numbers are the most natural, apparently 0 not in natural, 0 in whole
\(\mathbb{z}\) for integers, counting in german
rational numbers: a/b \(a,b \mathbb{e} \mathbb{z}\)
real numbers: infinite all the way down way more real numbers than rational numbers
- Zero: important for groups – starting point on number lines. true
neutral, Additive Identity
- Multiplicative Identity: 1
- identity lets it keep it's identity? when the op doesn't change
- negs: so we can deal with negs? so we can undo addition
subtraction is a lie! add negs subtraction on the natural numbers is not closed
closed: can't make a number not in the set
2 Groups
any set of mathematical elemements under one operation such that there is an identity each element has an inverse
- they do not need to be communitive
- a+b = b+a
- associativity
- (a+b)+c=a+(b+c)
- order doesnt matter
- most things we are doing will be associative
- nice number systems are almost always associative
can add dimensions, like complex adding more leads to quaternions or hamiltonians, then to sadonians?
called the cayley dickson construction, or smt
2.0.1 axioms:
- there exists an identity
- each element has an inverse
- it's closed
- associativity
3 Matrices
- can be called an array
- 2d can use rows and columns as coords
3.0.1 operations:
addition: only if same dimensions, loop through indicies dot: cross: wrong! first row by first column with addition to make first entry, first row by second column for second entry loop through indicies like addition
3.0.2 vectors:
special case of matrix
column vec ( 1, 2 )
row vec ( 1, 2 )
cannot add diff dimensions
3.0.3 representations
can draw up to 3, ish geometric is just arrow on graph to coords
adding vecs on the graph is just tip to tail, then first tip to last tail for resultant just like phys
(
a1
a2
.
.
.
an
)
is a vector of \(\mathbb{r}^n\)
matrix multiplication identity? multiplication on group? multiplication on to collum vectors
Homework: - KBxGroupAndMatricesIntro