Probability/Stats Definitions

Probability facts/definitions/theorems:
* Law of Total Probability: P[Y] = P[Y|X = 1]P[X = 1] + P[Y|X = 0]P[X = 0] . More generally for n possible discrete values of X: P[Y] = \sum_n P[Y|X_n] P[X_n]
Linear Algebra facts/definitions/theorems: 
* Rank of a Matrix: the maximum number of linearly independent columns. The dimension of the vector space generated/spanned by its columns.
* Full rank: a matrix X is said to have full rank if rank(X) = min\{\#rows, \#columns\}
* The column rank and row rank of a matrix are equal: for any given matrix X, rank_{row}(X) = rank_{column}(X)
*A non-square matrix is never invertible. 
*Invertible/Nonsingular/Nondegenerate Matrix: for some n\text{-by-}n square matrixX over a field K, these statements are equivalent: 
**There exists a matrix Y such that XY = I_n = YX
**X has full rank. That is, n = \text{rank}(X).
**The columns of X are linearly independent
**The columns of X span K^n
**The columns of X form a basis for K^n
**0 is not an eigenvalue of X
**The transpose X^\topis an invertible matrix