Transcription of Lecture 3 Linear Equations and Matrices
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Lecture 3 Linear Equations and Matrices
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-vectoryi
so multiplication by matrix inverse solves a set of linear equations some comments: • x = A−1b makes solving set of 100 linear equations in 100 variables look simple, but the notation is hiding alot of work!
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