Question

2 3 12 3 37 1. Let A - 10 15 40 7 1131 2 3 7 2 2 and B 1-2 -3 8 3 171 echelon form of A. (Assume this!) (a) (2 pt) What is the value of rank(A)? 110057 100 100 000121The B is the reduced to loooool (b) (2 pt) What is the value of nullity(AT)? (Read carefully (C) (3 pt) Find a basis for col(A). Circle your final answer. (d) (3 pt) Find a basis for row(A). Circle your final answer. (e) (4 pt) Find a basis for null(A). Circle your final answer.

Answer 1

a Trem that matrix and its reduce row pchlon form First I tell you something about rankf hallity of a matrix It A is any mxn matrix then rank-nullity theorm states that rank (A) + Nullity CA) = n . here n= coloumn of a matrix. Ten Where rank(A) = Number of linearly independent a row or coloumn of a matrix - Dim ( Now (A) or Dim (Col (A)). and Nullity CA = dim (Null (A)). Where Null CA) = 4X EROTAX=0} observe that in B last row is zero pee In matrix o last row of A are. linear combination of remaining row of A. This first three row of A are linearly Independ rankCA) = 3. S A is 445 matrix rankCA) + Nality CA) = 5 - Nullity CA) = 2 Also AT is 584 matrix. = rank CAT + Nullity CAT = 4 Ale have seen the det? of rank of a matrix vankCA) = No. of L. I row or coloumn in A. AS AN B then AT N BT a last coloumn of BT is zero. & remaining three coloumn of AT are L.I. & rank (AT) = 3. Nallity CAT) = 4-3 = 1

2 Pro observe entry) that in leading 1st, 3rd entry and Cire oth first non. coloumn in B. ist, 3rd and oth column of the basis of column space of ie Basis COLCA) 5121 12 7 3 A are An 17 ON A W In B A Basis first three row are L.I. of rOWCA) S (2, 3, 12, 3, 37), 1 ( 10, 15, 40,7, 113) 1 2 3 7 2 292 10 Basis BX of I 0 Null CA) is. 02 (2 ooooo d + Z up + 50g = 0 9 ble = 0 20+ gas O This is the system of three ego in tive variables, each ego make on variable constant dependent & remaing two variable are there. Let your fa ut are the variable.

5 + 9 lg = 0 x zgxs =-gr x + 2x + 50g = 0. = x1 = -3 t - 50. rite - ca au 24 & Basis of NallCA) = 5 LO