Question

4.4.16. Consider two matrices Ayxn and Bmxk. (a) Explain why rank (A | B) = rank (A) +rank (B) - dim (R(A) R(B)). Hint: Recall Exercise 4.2.9. (b) Now explain why dim N ( A B) = dim N (A)+dim N (B)+dim (RA) R (B)). (c) Determine dim (R(C) ON (C)) and dim (R(C) + N (C)) for -1 1 1 -2 1 -1 0 3 -4 2 C= -1 0 3 -5 3 -1 0 3 -6 4 1-1 0 3 -6 41

Answer 1

Let Me obe Ma (mi)rxa the rank (M) = dim RCM) =an{ MA E R/XERS} -Humn ? (1)-(1)-(3) (: RS has basi -dim = din (span of columns of M.) rank (A1B) = dim (x+W)/span of column of AlB) where = span of columns of A, We n B a span of columes of (AIB) a skam of columns of A and B = span {c..., k} (where cd is gth columnes of A o is ith column of 2) esprease of a a span { cha .... casa span {es,... 8} = V + W i rank (AIB) a dim IV+W). Let unwa ve ever he has a basis :{...... } we extend this to a basis of V: {v, ..... VE ... } : dim (v) = rank (A) = l, dim (W): rank (B)- ttm. Vtw = span {......, 4. ve www... wm} and this is a basis of VtW ( these are linearly independent has the

In fast saivi W; no fon ai, b; sonders => byloje (43) EVOW span {onales » Et; V; PER:) Vi ficken for tem sonhar ca But {v..., Vit, W, wms are lively independent of Every Wess } are also Thun Cao for sket, b; to for lesen Cutte = are o for lekst f ago for talegs e => {... ... ... wm is a line ind, set streton spanning Vtwo rank (A/B)= dim (V+W) = l+m = ltltim) - t = lim Vadim Wrain Now =YankA+rank (B) - Liw (A) ARG) 0 (AIB) is mx (kth) matris ) is mx (kin) matrix, then it operates from R ** -> RM dim N (AIB) = nek - Rank (A1B) (& rank nullity then) Similarly, dim - n - Rank (A), Linn NCB) K-Rank (B). i dim N CAHB) ntk - Yank A) - varte (13) + dims RCADOR()) ={n-rank (A)} + Ske-rande (81) + dim(CA)R@)) - dim N(A) + dim NCB) +timRIA) ARCB)) ® Apply now operation on to find RREF of c: -I -31 P RE-R

- 0 3 - 4 21 -2 2 -1 +1 Re'n-R2 R~RL +22 R ~ R; R 2 2 oo-2 =3,4/ Ī NON I 0 - 3 4 - 2 Ri~-RI R3'~.R3 Run R4 + 2831 -O. rowbank of c is 3 (1:3 independent rows) 7 din RCC) a vandre) - resa dimlow skar) - 3 les c is a 5x5 madrix, dim NCC) = 5-rank C=1. Abso, NCC)- ŞXERS | 6x = 0;} Foron ke BF 1 c -37, +46-2*5 = 0 : X(), *-1*3 +224 - uçao 24 - XS-O >? =25 > X2=293 -75 24 -323 - 25 Put X3=0, 2x *4=15 ₂x, = 32-2M, X2=2x-M i. N(C)= skan ? TO- Apply colum operations on C to find a spanning set of Roc)= spare spannende column of C 13. Inc; -2.C,C,,j=1,3,5,5 ) 3 -5 3 3 - OTī ii OOO-OOO-. ---- ooo..... -1 -2 -2 + +2 +2 / ~ Cu +2c, 0 O- 0 0 O 0 O - cres can cut 5 Red=span * exe-m 090) 2

e ReciNed - • (1-0)-(0) - 400 a . - OOOO => 3 einlebeeldent God Mount rank if the coefficient matrix is 3. my hande of reell space of four . 63e2 sor" sponce & is of na dimension 2 :dim (RCC) AN (6)) = 2 = dim Nees i NCO CROC) > RCC)-NCUS RCC). .: din (RCC) AN(O)) = 2 dim (Ro+Nco ) = din & cet & vane & =3.