M2-3C(Matrices)

def: matrix

an mxn matrix A is a rectangular array of elements of F with m rows and n columns

def: matrix of a linear map

let T be a linear map from V to W and v1,…,vn is a basis of V and w1,…,wm is a basis of W. the matrix of T is the mxn matrix with entries defined by Tvk=A1,kw1+…+Am,kwm

def: matrix addition

the sum of two matrices of the same size is the matrix obtained by adding corresponding entries of both matrices

thm: matrix of the sum of linear maps

let S and T be linear maps from V to W then M(S+T)=M(S)+M(T)

def: scalar multiplication of a matrix

the product of a scalar and a matrix is the matrix obtained by multiplying each entry by the scalar

thm: the matrix of a scalar times a linear map

let a be a scalar and T be a linear map from V to W then M(aT)=aM(T)

thm: dimFm,n=mn

with addition and scalar multiplication defined as above, Fm,n is a vector space of dimension mn

def: matrix multiplication

let A be an mxn matric and B an nxp matrix then AB is the mxp matrix whose entry in row j column k is computed by multiplying corresponding entries of row j of A and column k of B and then summing them

thm: matrix of product of linear maps

if T is a linear map from U to V and S is a linear map from V to W then M(ST)=M(S)M(T)

notation: Aj,.

the 1xn matrix of row j of A

notation: A.,k

the mx1 matrix of column k of A

thm: entry of matrix product equals row times column

A is an mxn matrix and B is an nxp matrix then the entry in row j column k of AB is row j of A multiplied by column k of B

thm: column of matrix product equals matrix times column

A is an mxn matrix and B is an nxp matrix then column k of AB equals A times column k of B

thm: linear combination of columns

Suppose A is an mxn matrix and b is an nx1 matrix then Ab is a linear combination of the columns of A with the scalars from b

thm: matrix multiplication as linear combination of columns

C is an mxc matrix and R is a cxn matrix then column k of CR is a linear combination of the columns of C with coefficients from column k of R

thm: matrix multiplication as linear combination of rows

C is an mxc matrix and R is a cxn matrix then row j of CR is a linear combination of the rows of R with coefficients from row j of C

def: column rank

dimension of the span of the columns of A

def: row rank

dimension of the span of the rows of A

def: transpose

let A be an mxn matrix then At is the nxm matrix made by interchanging the rows and columns of A

thm: column-row factorization

A is an mxn matrix with column rank c greater/equal to 1 then there is an mxc matrix C and cxn matrix R such that A=CR

thm: column rank equals row rank

Suppose A is an mxn matrix then column rank of A is equal to the row rank

def: rank

rank of an mxn matrix is the column rank of that matrix