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ALA240-2009 - November 24th

ALA240-2009 - November 24th - November 24th 2009 MAT240...

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November 24 th , 2009 MAT240: Abstract Linear Algebra Lecture: Apparently, the next class will be awful. In general, variables ? 1 … ? ? ? ?? ∈ ? ? 1 … ? ? ∈ ? ? 11 ? 1 + ? 12 ? 2 + + ? 1 ? ? ? = ? 1 ? 21 ? 1 + ? 22 ? 2 + + ? 2 ? ? ? = ? 2 ? ? 1 ? 1 + ? ? 2 ? 2 + + ? ?? ? ? = ? ? 𝐴 = ? 11 ? 1 ? ? ? 1 ? ?? , ? = ? 1 ? ? , (unknown vector), ? = ? 1 ? ? ∈ ? ? → 𝐴? = ? Solve for the coordinates of unknown vector. If ? = 0 → 𝐴? = 0 homogeneous If ? ≠ 0 → 𝐴? = ? inhomogeneous If lucky, A is invertible, i.e, there is 𝐴 1 ? . ? . 𝐴 1 𝐴 = 𝐼 𝐴? = ? → 𝐴 1 𝐴 = 𝐴 1 ? → ? = 𝐴 1 ? So equation has a unique solution and it is ? = 𝐴 1 ? Observations regarding the homogeneous case 𝐴? = 0 : 1. 𝐴? = 0 ↔ 𝑇 𝐴 ? = 0 ? ∈ ????????? 𝑇 𝐴 = 𝑁 𝑇 𝐴 = 𝐾?? 𝑇 𝐴 = 𝑁 𝐴 = 𝐾?? ( 𝐴 ) 2. The set of solutions in 𝑁 𝐴 , is a subspace of ? ? . ????????? ???????? = ??????? ( 𝐴 ) 3. 0 is always a solution.
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Observations regarding the inhomogeneous case 𝐴? = ? : 1. Solutions exist iff ? ∈ 𝑅 𝐴 = ???𝑔? 𝐴 = ?????? − ???? 𝐴 2. If ? 0 is a solution of 𝐴? = ? , then ? 1 is a solution too if ?
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