Lecture 31 Note Linear transformations are not on the midterm Linear transformations T ax by aT x bT y T 0 0 T input output Matrix multiplication is a linear transformation A ax by aAx bAy T x is the projection of x into the plane p All linear transformations have a matrix hiding behind them Introduce a basis for R then we get a matrix P Example take the standard basis in R But we can choose a different basis Let the eigenvectors be the basis Now A Linear Transformation Steps Choose transformation T Choose a basis for R Choose a basis for R Construct the matrix A Shift is not a linear transformation Rescaling is a linear transformation Rotating is a linear transformation Example Let input and output spaces be 2x2 Dimension of the input space and the output space are 4 Matrix M has to be a 4x4 matrix A basis for all 2x2 matrices is
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