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Unformatted text preview: 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 + ⋯ + ? ¥? ? ? = ¤ ¥ = ? 11 ⋯ ? 1 ? ⋮ ⋱ ⋮ ? ¥ 1 ⋯ ? ¥? ¦ , ? = ? 1 ⋮ ? ? ¦ , (unknown vector), ¤ = ¤ 1 ⋮ ¤ ¥ ¦ ∈ £ ¥ → ? = ¤ Solve for the coordinates of unknown vector. If ¤ = 0 → ? = 0 homogeneous If ¤ ≠ → ? = ¤ inhomogeneous If lucky, A is invertible, i.e, there is − 1 § . ¨ . − 1 = ? = ¤ → − 1 = − 1 ¤ → ? = − 1 ¤ So equation has a unique solution and it is ? = − 1 ¤ Observations regarding the homogeneous case ©? = 0 ª : 1. ? = 0 ↔ ©?ª = 0 ? ∈ ??««§??¬© ª = © ª = ®© ª = ©ª = ®...
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This note was uploaded on 05/25/2011 for the course MAT 244 taught by Professor Christinasaleh during the Fall '10 term at University of Toronto.
 Fall '10
 ChristinaSaleh
 Linear Algebra, Algebra

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