Lecture 33 Linear Transformation T c v c v c T v c T v If we know know T v T v and if v v a basis for the input means that every vector is one combination of the v s If we know T v for v basis vectors v v Then we know T v for all v Next When is T invertible Know that T v w where j 1 n w w must be a basis for the output space A v v w w AV W when is A invertible We know V is invertible A WV therefore A is invertible when W is invertible m x n matrix r rank output space dim col space r we know T x Ax m not necessarily m unique columns dim nullspace n r dim column space dim nullspace dim input space dim input space r Every linear transformation T can be described as a matrix T v x v x linear because it is linear in v input space v x a a x a x basis input and output 1 x x dim input space 3 output space apply basis T 1 0 T x x T x 2x notice that it is like eigenvectors with eigenvalues 0 1 2 matrix same as dim outputs col space range 2 dim nullspace kernel 1 dim input space 3 Thus T v x v x 0 a x 2a x Change of Basis Finding the matrix need to know T need to choose input basis and output basis Example T v v identity transformation input basis v and v output basis matrix for T Rule apply T to each basis vector v v know T v T v Now same T identity input basis v and v output basis Column 1 comes from T v a w a w a a
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