LectureNotes_7.3and7.4.pdf

# The rank dimension theorem recall theorem 222 if o

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The Rank (Dimension) Theorem Recall: Theorem (2.2.2) If o homogeneous l'inear system has n unknou)ns. and i,f the reduced row echelon form of i,ts augnxented matrir has r nonzero rou)s, then the system Itas r leadr,ng uariables and rL - r free uariables. Suppose A is an n'L x n rnatrix, B is a RREF of A. We have: frt uieJ) # of leading variables of -R # of nonzero rows of ft # of free variables of E # of parameters in general solution toAx-0 size of a basis for null(A) # of columns ofR n rL + + size of a basis for + row(A) - row(B) 4 dim(ro*(A)) + dim(null(A)) TL (

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The ( Rank (Dimension) Theorem co!,-!-'- TE* in CJ^.} . ( T+A .o*f{. <-n*Sr,"1 of c*,s\ler.l s3s-{<r,n, +h"^ (# 4 f.* u..ns)= n - I-an6 CA). For any matrix A, tlne rows of A are the column s of AT , and therefore we get that rank (A' ) - rank(A). Theorem (The Rank( Dimension) Theorem) Let A be a,n m x n matrir. Then rank(A) * nullity (A) - n nu-LLi 5 + fl.ovt &b \??d\ R CF. Sorne consequences of the Rank Theorem: Theorem (7.4.2) K QP n=f'ro rf A'isanmxnn'L w?, t ank(A) k then: \< Z?-r]c-' {Q-P5 ' (u) nullity (A) tvtvnorls' (A)= h- rank cA) rfi l I , (b) Euery REF of A has k nonzero rows an (.) The homogeneous system Ax - 0 has k uariables. ero Tows. 5 +G-r):\1 n- m-k gua riables andn-k free
U\$ C{ Ra""A />inrte^sian TR.) L"f k be o,^,^ fn x n ,l-y\ttT ix R RRctr 4 A -)Kh 1- coLmn-g c^nheAt5 -) ra^^.}( CA) =F. r) Va,\i o,Dea + l-{,'- *--\e*d;6 L flG^ A t'-d ,^rgR^-o* U,T A,l.s \$;s"- r*).ri'f6. coLrmru + R" Uv\Q- Wvcnr.aiAr4, + "*ItB CA) - (\- r. ) ( \\,^4, ca,*k LA) + "Jt',5 CA) -- r + Cn-O : n' 6 O^J> r 4 d t {-.J

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'fhe R ank Theorem ( Example Find the rank and nullity of the following matrices , -J 3 2 4x6 A B 1 5 0 5 4 5 0 5 1 3 1 , J 5 5 5 4 0 5 5 -5 4 3 1 0 4 5 5 5 -3 4 -4 -4 -2 2 -1 -5 -5 0 4 -5 o -J a, -J 0 a, -e) -1 2 5 -2 -5 2 4 I trt\trno-r n . is a 4x 7 matrix. Can Ahave a out a four-dimensional null space Y\d,l;br\$ =Q y* \ qe\&lns/*"-,'[email protected] 10x3 Erample.
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