In: Math
Let S be the set of vectors in R4
S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)}
(a) State whether the vectors in the set are linearly
independent.
(b) Find the basis for S.
(c) Find the dimension of the column space (rank S).
(d) Find the dimension of the null space (nullity S).
(e) Find the basis for the null space of S?
Question-(a)
.
write all vectors in the matrix
there is no pivot entry at last column so vectors are NOT linearly independent
.
.
.
Question-(b)
.
here we have 3 pivot entries at first 3 columns so the basis for S is
.
.
Question-(c)
.
here we have 3 pivot entries so the dimension of the column space is 3
.
.
Question-(d)
.
there is no pivot entry at last column so the dimension of the null space is 1
.
.
Question-(e)
.
reduced system is
general solution is
the basis for null space is