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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...

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?

Solutions

Expert Solution

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

milcah answered 1 year ago

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