Transcription of Linear algebra cheat-sheet - Laurent Lessard
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Linear algebra cheat-sheet - Laurent Lessard
Linear algebra cheat-sheetLaurent LessardUniversity of Wisconsin MadisonLast updated: October 12, 2016 Matrix basicsA matrix is an array of Rm nmeans that:A= (mrows andncolumns)Two matrices can be multiplied if inner dimensions agree:C(m p)=A(m n)B(n p)wherecij=n k=1aikbkjTranspose: The transpose operatorATswaps rows andcolumns. IfA Rm nthenAT Rn mand (AT)ij=Aji. (AT)T=A. (AB)T= basics (cont d)Vector products. Ifx,y Rnare column vectors, Theinner productisxTy R( dot product) Theouter productisxyT Rn are just ordinary matrix multiplications!Inverse. LetA Rn n(square). If there existsB Rn nwithAB=IorBA=I(if one holds, then the other holds with thesameB) thenBis called theinverseofA, denotedB=A properties of the matrix inverse: A 1is unique if it exists.
Oct 12, 2016 · Linear equations Given A 2Rm n and b 2Rm, linear equations take the form Ax = b Where we must solve for x 2Rn.Three possibilities: No solutions. Example: x 1 + 2 = 1 and 1 2 = 0. Exactly one solution. Example: x 1 = 1 and 2 = 0. In nitely many solutions.
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