Question

Find bases for the four fundamental subspaces of the matrix A as follows. N(A) = nullspace of A N(AT) = nullspace of AT R(A) = column space of A R(AT) = column space of AT Then show that N(A) = R(A) and N(AT) = R(A)". 1 1 0 0 2-3 -1 1-3 N(A) = 11 N(AT) 11 R(A) 11 R(A) = 3 1

Answer 1

A = 0 2-3 R3TR 2 -3 1 2 P2 0 1 -3/2 2 - 3 R3-2R2 O -312 let x= E Nallea) Then nithgo 49-32=0 - x=-2g 13 -ng -3 3 202/2 so Nell CA) = Span

0 1 Now A = 2 0 -3 -3 P, P, T 0 2 2 0 -3 -3 ÍR 0 -3 -3 R2 + 3 R2 0 0 O x- natrz 20 ca az and y = -xz

93 BE x2 so NCAT) = Spen SED column space of AT es RCAT) = Span column space of A is RCA) : z span 010 y E RCAT Then 8,-8, = 0 4,+262 +2=0 y = 93 2 = -2 yo so RCA) - - Span SED (er Care Then y +42=0 29-30 320 y = -999 3 202 ܟ ܟ ܝ *2 y2 2843 28 So RCA&T) - - Span 3 Therefore N(A) = R(AT) RCAT + and N(AT) = R(A) 2