Question

Assume that the matrix A is row equivalent to B. Without calculations, list rank A and dim Nul A. Then find bases for Col A, Row A, and Nul A. 1 3 -5 -7 2 1 3 -5 - 7 N -2 -6 12 16 -9 0 0 1 1-5 A= B = 2 6 -16 - 20 34 0 0 0 0 5 -3 -9 6 12 0 0 0 0 0 0 rank A= dim Nul A= A basis for Col Ais {C}. (Use a comma to separate vectors as needed.)

Answer 1

# Assume that the Matrix A is now equivalent to B. without calculation, lisk ranka and din NWA. then find bases for Col A Rowa, and nulla 3 -5 A= - 0 0 -2 -6 2 6 -5 -7 2 12 169 Be -16 -20 34 6 12 0 '485 onú 0 o 0 © o [-3 -9 485 Solution- Rank Nullity theorems : het velf) be a finite dimensional vectoy space and T: v(IF) → ULIF) dim (N(T)) + din(R(T) = din of domain space be LT then zė In(t) + 8(T)=n where nis dimension of Vector Space VIF) Now Here In the given Question A is now equivalent to Matrix B and -63 -7 Ba 0 -5 0 0 0 0 O 0 o 4XS le we have 3 pivat hence Rank (B) = 3 and since B is equivalent to a zë Ramie A=3 and since 4X5 5 Az (0exis TO IR and By Rank, Nunity theorem ܟܐ IR

dim N(T) + din R(T) = din 01 domain space Të din N(T) + 3=5 re din N(T) 25-3=2 e 'dim N(T)=2 2 ? Mula = 2 Now We find bases for cola, Row A and Nue A basis of NULA I layo TEIREIRA Ax=0 where Az -5 -7 2 3 -2 -6 12 16 -9 R 0 2 f 1-3 -9 -16 6 4 o -20 34 12 0 405 us 5*1 tierien Its Echelon form] 13 -s -7 au hr 0 1 sooo 2 0 0 n4 H x, +3X₂ - Saz-7X4 + 2X620 Xz+4 -5520 > (+2420 54 20 = ingzo 9 kg =-24

Basis, IB, 0 0 this is basis for Nulla O ܐ T T 0 0 O Basis for Row A do Row space is subspace of domain space since Given B is row Equivalent to a -7 where BE 3 -5 2 I s 0 O - C o 0 5 0 o O O 0 1485 b's Trauspose of B= O 0 0 3 o o -5 ~) 0 5 2 -5 5x4 & Basis for Row A 6 o B2= نما 0 -5 -7 2 0 5

© Basis for cal Aro Given B2 3 -5 -7 2 4 ů TEIRS IR 0 o 0 o 0 o 0 O o 4x5 2 Basis B, - -5 o Cal A is subspace of Co-demain space ů Coloumm space is subspace of co-demain space