week1_605

#install.packages('pracma')
library('pracma')

C26 pg. 255

D = cbind(c(-7,6,3,-1), c(-11,10,5,-2), c(-19,18,9,-4), c(-15,14,7,-3))
I = diag(4)
EEF = cbind(D,I)
EEF

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]   -7  -11  -19  -15    1    0    0    0
## [2,]    6   10   18   14    0    1    0    0
## [3,]    3    5    9    7    0    0    1    0
## [4,]   -1   -2   -4   -3    0    0    0    1

RREEF = rref(EEF)
RREEF

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    1    0   -2   -1    0    0    2    5
## [2,]    0    1    3    2    0    0   -1   -3
## [3,]    0    0    0    0    1    0    3    2
## [4,]    0    0    0    0    0    1   -2    0

1. A linearly independent set whose span is the column space of D. Colspace of D is Nullspace of L; C(D) is N(L)

L = RREEF[3:4,5:8]
L

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    3    2
## [2,]    0    1   -2    0

null_L = cbind(c(-3,2,1,0), c(-2,0,0,1))
null_L

##      [,1] [,2]
## [1,]   -3   -2
## [2,]    2    0
## [3,]    1    0
## [4,]    0    1

L%*%null_L

##               [,1]          [,2]
## [1,] -8.881784e-16 -1.776357e-15
## [2,]  0.000000e+00  0.000000e+00

The columns of null_L are a linearly independent set whose span is the columns space of D

2. A linearly independent set whose span is the left null space of D. Leftnullspace of D is rowspace of L; LN(D) is R(L) Rowspace of L is colspace of L-transpose; R(L) is C(L-t)

Lt = t(L)
Lt

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1
## [3,]    3   -2
## [4,]    2    0

The columns of Lt are a linearly independent set whose span is the left null space of D