the symbol R refers to all real numbers, also known as scalars

the symbol ∈ is used to indicate membership

vector in R^n is a list of n scalars organized vertically into a list (lower-case bold letters)

MATRICES

an m x n matrix is an array of scalars with m rows and n columns

• matrix A ∈ R^(m x n)

• n x n matrix is a square

the (i,j) entry of a matrix is the scalar in the ith row and jth column

• (row, column)

scalar-vector products work coordinate-wise

vector addition also works coordinate-wise

• only works if coordinates are the same

scalar-matrix products work component-wise (multiply every entry by scalar)

matrix addition also works component-wise

ONLY VECTORS AND MATRICES WITH THE SAME SHAPE CAN BE SUMMED

OPERATIONS

the transpose A^T is formed by interchanging the rows and columns of A

• the (i,j) entry of A^T is a_ji

• A^T is n x m if A is m x n

• transposing saves vertical space by writing vectors as transposes of 1 x n matrices

  • horizontal notation

transposition is a linear operation:

• scalar matrices and transposing can be distributed

  • c(A + B)^T = cA^T + cB^T

• transposition is an involution:

  • (A^T)T = A

a matrix is symmetric:

• S^T = S

the trace of an x n matrix is the sum of its diagonal

• trace is linear operation, you can distribute it