Basis

Spanning Sets and Bases

Basis and Dimension

Recall Null and Row Spaces of A

library(pracma)
library(MASS)
A <-matrix(c(1,-1,4,2,0,-1,-1,-1,5),nrow=3,ncol=3,byrow=TRUE)
rref(A)

     [,1] [,2] [,3]
[1,]    1    0 -0.5
[2,]    0    1 -4.5
[3,]    0    0  0.0

Null(t(A))

          [,1]
[1,] 0.1078328
[2,] 0.9704950
[3,] 0.2156655

Rank(A)

[1] 2

What is the dimension of the null space of A? What is the dimension of the row space of A?

What is the Rank-Nullity Theorem? (see Rank–nullity theorem - Wikipedia)

Exercise