Introduction to Groups and Matrices

I spend most of my free time doing programming projects with my friends, and recently I've been doing a lot of Machine Learning. My 'excuse' for taking linear algebra is that in my more recent ML projects I've had to go lower level and I'm being held back by my understanding of linear algebra and statistics, but frankly linear algebra just sounds really cool and I enjoy Nueva math classes a lot.

monday! *

:CUSTOM_ID: read-1.a-from-the-textbook.-we-will-discuss-any-questions-on-monday

q: is division also a lie? yes! q: do tuples all need the same type? why call them n-tuples instead of lists? q: why is it called liniear algebra? the explanation that it doesnt deal with geo doesnt explain it

Under multiplication?*

:CUSTOM_ID: which-of-the-number-systems-we-discussed-are-groups-under-addition-under-multiplication

• natural
  • + N: no identity
  • * N: no inverse
• whole
  • + N: no inverse
  • * N: no inverse
• integers
  • + Y
  • * N: no inverse
• rational
  • + Y
  • * Y : wrong! no inverse for 0. ish…
• real
  • + Y
  • * Y
• complex
  • + Y
  • * Y missing zero!

group requirements?*

:CUSTOM_ID: is-there-a-multiplication-for-3-by-1-vectors-that-satisfies-all-group-requirements

Y – Multiply equal indices.

E.g. $

\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}

$ $

\begin{bmatrix} 2 & 2 & 2 \end{bmatrix}

=

\begin{bmatrix} 1*2 & 2*2 & 3*2 \end{bmatrix}

=

\begin{bmatrix} 2 & 4 & 6 \end{bmatrix}

$

Y – $

\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

$

0s cancel the multiplications where the operation index doesn't match the sum of the matrix indices.

expand to 1s along the diagonal