"File: GRAM"

"###################################"

"AUXILIARY FUNCTIONS"

U_(z):=z/ABS(z)

D_(x):=DIMENSION(x)

E_(z,i):=ELEMENT(z,i)

HELP_G(z,w,k):=VECTOR(IF(i<k,E_(z,i),U_(E_(w,k)-SUM((E_(w,k) . E_(z,i))*E_(z,i~
),i,1,k-1))),i,k)

GRAM_H(z,w,k):=IF(k>D_(w),z,GRAM_H(HELP_G(z,w,k),w,k+1))

"########################################"

"USER FUNCITONS"

GRAM(w):=GRAM_H([U_(E_(w,1))],w,2)

"#################################################"

"INSTRUCTIONS"

"If w is a matrix whose rows are independant"

"then GRAM(w) returns a matrix whose rows are an orthonormal basis"

"for the space spanned by the rows of w."

"########################################"

"EXAMPLE"

a:=[[2,4,1,0,0],[2,3,4,3,6],[4,2,3,1,1]]

"Simplify the following to see Gram-Schmidt applied to a"

GRAM(a)