MATA23H3 Lecture Notes - Lecture 16: Invertible Matrix, Linear Map, Free Variables And Bound Variables
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Mata23 - lecture 16 - linear transformations and determinates. One-to-one and onto: let t : rn rm be a linear transformation. T is one-to-one if and only if ker(t ) = is onto if and only if range(t ) = rm (see section 2. 1, #38) (cid:111) (cid:110)(cid:126)0. Given t is one-to-one, want to show: * ker(t ) = (cid:111) (cid:110)(cid:126)0 (cid:110) (cid:126)x|t ((cid:126)x) = (cid:126)0(cid:48)(cid:111) Let t ((cid:126)u) = t ((cid:126)v) (cid:126)u, (cid:126)v rn. T ((cid:126)u) t ((cid:126)v) = t ((cid:126)u (cid:126)v) = (cid:126)0(cid:48) (cid:126)u (cid:126)v ker(t ) (cid:126)u (cid:126)v = (cid:126)0 = (cid:126)u = (cid:126)v. T is one-to-one (cid:110)(cid:126)0 (cid:111) (cid:111) (cid:110)(cid:126)0. Given t is onto, want to show range(t ) = rm. (cid:126)y rm, (cid:126)x rm such that t ((cid:126)x) = (cid:126)y. Given range(t ) = rm, want to show t is onto. (cid:126)y rm, (cid:126)y range(t ) (cid:126)x rn such that (cid:126)y = t ((cid:126)x) Subspace w rn, dimw = n = w = rn.
