2.4 Invertibility 1

2.4 Invertibility 1 - 2.4. 2.4. Invertibility and...

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2.4. 2.4. Invertibility and Isomorphisms 1. (a) F, ([ T ] β α ) - 1 = [ T - 1 ] α β (b) T (c) F A Mat m × n ( F ) , L A : F n F m [ L A ] β α = A in case of α, β are the standard bases (d) F, dim ( M 2 × 3 ( F )) 6 = F 5 (e) P n ( F ) P m ( F ) iff n=m ( ) clear ( ) T : P n ( F ) P m ( F ) is isomorphic Since T is one-to-one and onto, dim ( P n ( F )) = rank ( T ) + nullity ( T ) = dim ( P m ( F )) n + 1 = m + 1 n = m (f) F (In case A and B are n × n matrices, it’s true) (g) T (h) T (Exercise 8) (i) T 2. T is invertible iff [ T ] γ β 101 PNU-MATH
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2.4. (a) T : R 2 R 3 , T ( a 1 ,a 2 ) = ( a 1 - 2 a 2 ,a 2 , 3 a 1 + 4 a 2 ) For β = { (1 , 0) , (0 , 1) } and γ = { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) } bases for R 2 ,R 3 , re- spectively T (1 , 0) = (1 , 0 , 3) = 1 · (1 , 0 , 0) + 0 · (0 , 1 , 0) + 3 · (0 , 0 , 1) T (0 , 1) = ( - 2 , 1 , 4) = - 2 · (1 , 0 , 0) + 1 · (0 , 1 , 0) + 4 · (0 , 0 , 1) [ T ] γ β = 1 - 2 0 1 3 4 is not a aquare matrix
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2.4 Invertibility 1 - 2.4. 2.4. Invertibility and...

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