Transcription of Lecture3 LinearEquationsandMatrices
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Lecture 3 linear equations and matrices linear functions linear equations solving linear equations3 1 linear 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 2 Matrix 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 3 Composition 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 4 linear 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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Linear algebra, Matrices, Linear equations, Linear, For Linear Systems of Differential Equations, Linear systems of differential equations, Equations, Inverse matrix to solve equations, Solving Simultaneous Equations and Matrices, Introduction to Linear Algebra, Linear Equations and Matrices, Linear Algebra I - Lectures Notes - Spring