Lecture3 LinearEquationsandMatrices

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Lecture 3Linear equations and matrices linear functions linear equations solving linear equations3 1Linear functionsfunctionfmapsn-vectors intom-vectors islinearif it satisfies: scaling: for anyn-vectorx, any scalar ,f( x) = f(x) superposition: for anyn-vectorsuandv,f(u+v) =f(u) +f(v)example:f(x) =y, wherex= x1x2x3 ,y= x3 2x13x1 2x2 let s check scaling property:f( x) = ( x3) 2( x1)3( x1) 2( x2) = x3 2x13x1 2x2 = f(x) linear equations and Matrices3 2Matrix multiplication and linear functionsgeneral example:f(x) =Ax, whereAism nmatrix scaling:f( x) =A( x) = Ax= f(x) superposition:f(u+v) =A(u+v) =Au+Av=f(u) +f(v)so, matrix multiplication is a linear functionconverse:everylinear functiony=f(x), withyanm-vector andxandn-vector, can be expressed asy=Axfor somem nmatrixAyou can get the coefficients ofAfromAij=yiwhenx=ejLinear equations and Matrices3 3Composition of linear functionssuppose m-vectoryis a linear function ofn-vectorx, ,y=AxwhereAism n p-vectorzis a linear function ofy, ,z=BywhereBisp a linear function ofx, andz=By= (BA)xsomatrix multiplicationcorresponds tocompositionof linear functions, , linear functions of linear functions of some variablesLinear equations and Matrices3 4Linear equationsan equation in the variablesx1.

Linear Equations and Matrices 3–9. when A isn’t invertible, i.e., inverse doesn’t exist, • one or more of the equations is redundant (i.e., can be obtained from the others) • the equations are inconsistent or contradictory (these facts are studied in linear algebra)

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