Class Note for MATH 790 at KU
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Date Created: 02/06/15
Transpose Linear Algebra Notes Satya Mandal September 227 2005 Let F be a led and V be vector space over F With dimV n lt 00 De nition 01 Let V7 W be two vector spaces over F and T V a W be a linear trancformation We de ne a map Tt W a V by de ning Ttf foT V a F forf E W Diagramatically V gtT W T U if F We say that Tt is the transpose of T Note that the transpose Tt is linear transformation Theorem 01 Let V7 W be two nite dimensional vector spaces over F with dimV n and dimW m Let TVaW be a linear transformation Assume 617 76 is a basis of V and 617 76m is a basis of W Suppose A is the matrix ofT with respect to these two bases Then the transpose At is the matrix of Tt with respect to the dual bases 6 ej n of W and e 76 of V 1 Proof We have T61 7T n 617 7EmA Write A aij Apply7 for example7 6f to the above equation7 we get fT 17 7 fT n 1707 70A a117 7a1n This means that 111 Wei 7 an em 041m Similarly7 for 239 17 7771 we work with 67 and get an Tier 7 a vex am Therefore7 Tt f7 7Tt 67 6At m 7 n Hence At is the matrix of Tt7 with respect to these dual bases So7 the proof is complete Theorem 02 Let V7 W be two nite dimensional vector spaces over l7 with dimV n and dimW m Let TV7W be a linear transformation 1 Let R TV be the range of T and N be the null space of the transpose Tt Then the null space N annR 2 rankT ranCTt 5 Also rangeTt annNT7 where NT is the null space of T Proof To prove 17 we have null space of Tt N f E W Ttf0fEWifOT0fEW5fR0 annR To prove 27 note that dimW dimR dimannR rankT dimN and dimW dimW ranCTt dimN Therefore rankT ranCTt and the proof of 2 is complete Proof of 3 is similar to that of Theorem 03 Let A be an m X n matrix with entries in F Then Row 7 rankA Column 7 rankA Proof Let T F 7 F be the linear transformation given by TX AX Observe that TlF the row space of A By above theorem dimTX rankT ranCTt So7 R0w7 rankA rankT ranCTt Row 7 ranCAt column 7 rankA Therefore the proof is complete Remark 01 One should try a direct proof of this theorem
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