linear algebra stuff works on things that are linear
basis vectors are the building blocks of vectors (it's like the squares in our 2d graph).
span is the space of the combinations that our vector can reach. sometimes your basis vectors can reach all available dimensions, sometimes not.
a matrix describes a transformation composed of its basis vectors. when you perform matrix-vector multiplication, you apply that transformation to the vector. matrix-matrix multiplication is just applying a transformation to another transformation. alternatively you can imagine matrix-vector multiplication as scaling the vector along the matrix's eigenvector by its eigenvalue (probably doesn't apply when eigenvalue is complex? idk???).
\(A\vec{v}=\lambda\vec{v}\)
\((A-\lambda I)\vec{v}=0\)
determinant describes how much your transformation (matrix) scales an area. Negative determinant means we invert ourselves, and a determinant of zero means we smash space to a lesser dimension (not necessarily one dimension down).
solve for \(\lambda\), \(det(A-\lambda I)=0\)
NOTE: discriminant is \(\sqrt{b^2-4ac}\)
Monoid: is a set, associative, has law of composition, and a unit element, represents a binary operation. e.g. \(S^2\mapsto S\)
\(\log_2(8) = x\) is asking us "what power do I take 2 to in order to get 8?"
\(\log_a(b) = c\) is equal to \(a^c = b\)
One time I was trying to find the amount of digits of a really large number like 11578902317. If the number had commas, it’d be easier to see the digit count, but it wasn’t formatted that way. We could use \(\log_a(b)\) tells us how many digits \(b\) has, given an \(a\) based numbering system.
\(floor(log_{10}(11578902317)) + 1 = 11\) digits.
Commutative is about the order of operands
Associate is about the order of operations
Transitivity is about relationships