Linear maps characteristics II
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Exercise:
Let V be a finite dimensional vector space and v_...v_nin V be a basis for V. Let w_...w_nin W be arbitrary vectors. Then exists ! a linear map T:Vlongrightarrow W s.t. Tv_iw_iquad forall leq ileq n.
Solution:
Proof of the existence of T Let vin V. We want to define Tv. Since v_...v_n is a basis exists a_...a_nin K s.t. va_v_+...+a_nv_n. We define Tv:a_w_+...+a_nw_n. Note that T is well defined because forall v the coeffs. a_...a_n of the linear combination a_v_+...+a_nv_n that gives v are unique. bf We claim that T is linear. Indeed let vv'in V. Longrightarrow a_...a_nin K s.t. va_v_+...+a_nv_n and exists a_'...a_n'in K s.t. v'a_'v_+...+a_n'v_n Longrightarrow v+v'a_+a_'v_+...+a_n+a_n'v_n Longrightarrow by our definition of T Tv+v'a_+a_'w_+...+a_n+a_n'w_n a_w_+...+a_nw_n+a_'w_+...+a_n'w_n Tv+Tv'. Also if alphain K then alpha valpha a_v_+...+alpha a_nv_n Longrightarrow Talpha valpha a_w_+...+alpha a_nw_n alphaa_w_+...+a_nw_nalpha Tv. This proves that T:Vlongrightarrow W is a linear map. bf We claim that Tv_iw_iquad forall leq ileq n. Indeed v_i v_+...+ v_i+...+ v_n Longrightarrow Tv_i w_+...+ w_i+...+ w_nw_i. This concludes the proof of the existence of T. Proof of the uniqueness of T Suppose TS:Vlongrightarrow W are two linear maps with Tv_iSv_iw_iquad forall leq ileq n. Let vin V. We have to prove that TvSv. Write va_v_+...+a_nv_n. TvTa_v_+...+a_nv_na_Tv_+...+a_nTv_n a_Sv_+...+a_nSv_n Sa_v_+...+a_nv_n Sv
Let V be a finite dimensional vector space and v_...v_nin V be a basis for V. Let w_...w_nin W be arbitrary vectors. Then exists ! a linear map T:Vlongrightarrow W s.t. Tv_iw_iquad forall leq ileq n.
Solution:
Proof of the existence of T Let vin V. We want to define Tv. Since v_...v_n is a basis exists a_...a_nin K s.t. va_v_+...+a_nv_n. We define Tv:a_w_+...+a_nw_n. Note that T is well defined because forall v the coeffs. a_...a_n of the linear combination a_v_+...+a_nv_n that gives v are unique. bf We claim that T is linear. Indeed let vv'in V. Longrightarrow a_...a_nin K s.t. va_v_+...+a_nv_n and exists a_'...a_n'in K s.t. v'a_'v_+...+a_n'v_n Longrightarrow v+v'a_+a_'v_+...+a_n+a_n'v_n Longrightarrow by our definition of T Tv+v'a_+a_'w_+...+a_n+a_n'w_n a_w_+...+a_nw_n+a_'w_+...+a_n'w_n Tv+Tv'. Also if alphain K then alpha valpha a_v_+...+alpha a_nv_n Longrightarrow Talpha valpha a_w_+...+alpha a_nw_n alphaa_w_+...+a_nw_nalpha Tv. This proves that T:Vlongrightarrow W is a linear map. bf We claim that Tv_iw_iquad forall leq ileq n. Indeed v_i v_+...+ v_i+...+ v_n Longrightarrow Tv_i w_+...+ w_i+...+ w_nw_i. This concludes the proof of the existence of T. Proof of the uniqueness of T Suppose TS:Vlongrightarrow W are two linear maps with Tv_iSv_iw_iquad forall leq ileq n. Let vin V. We have to prove that TvSv. Write va_v_+...+a_nv_n. TvTa_v_+...+a_nv_na_Tv_+...+a_nTv_n a_Sv_+...+a_nSv_n Sa_v_+...+a_nv_n Sv