Orthogonal projection characteristics
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Exercise:
P_U has the following properties: abcliste abc P_U is a linear map. abc textIm P_U U textKer P_U U^perp. abc forall vin V v-P_Uvin U^perp. abc forall uin U P_Uuu. abc If e_...e_r is any orthonormal basis for U then P_Uv _i^r langle ve_irangle e_i. tildeP_Uv from the previous proof. abcliste
Solution:
Proof. abcliste a-d follow from the general fact that if W_ W_ are vector subspaces of W and W_oplus W_W then the map p_W_:Wrightarrow W_ defined by P_W_ww_ where we write ww_+w_ is a unique way with w_in W_ w_in W_ is a linear map textIm P_W_W_ textKer P_W_W_ and forall win W we have w-P_W_win W_. So wP_W_w+w-P_W_w is the decoposition of w as ww_+w_. We also have P_W_wwquad forall win W_. For e write vleft _i^r langle v e_irangle e_iright + v-_i^r langle v e_irangle e_i *. Now left _i^r langle v e_irangle e_irighttildeP_Uv from the previous proof and we have also seen in that proof that v-_i^r langle v e_irangle e_iin U^perp. Longrightarrow * is the unique decomposition of v with respect to VUoplus U^perpLongrightarrow P_Uvleft _i^r langle v e_irangle e_irighttildeP_Uv. abcliste
P_U has the following properties: abcliste abc P_U is a linear map. abc textIm P_U U textKer P_U U^perp. abc forall vin V v-P_Uvin U^perp. abc forall uin U P_Uuu. abc If e_...e_r is any orthonormal basis for U then P_Uv _i^r langle ve_irangle e_i. tildeP_Uv from the previous proof. abcliste
Solution:
Proof. abcliste a-d follow from the general fact that if W_ W_ are vector subspaces of W and W_oplus W_W then the map p_W_:Wrightarrow W_ defined by P_W_ww_ where we write ww_+w_ is a unique way with w_in W_ w_in W_ is a linear map textIm P_W_W_ textKer P_W_W_ and forall win W we have w-P_W_win W_. So wP_W_w+w-P_W_w is the decoposition of w as ww_+w_. We also have P_W_wwquad forall win W_. For e write vleft _i^r langle v e_irangle e_iright + v-_i^r langle v e_irangle e_i *. Now left _i^r langle v e_irangle e_irighttildeP_Uv from the previous proof and we have also seen in that proof that v-_i^r langle v e_irangle e_iin U^perp. Longrightarrow * is the unique decomposition of v with respect to VUoplus U^perpLongrightarrow P_Uvleft _i^r langle v e_irangle e_irighttildeP_Uv. abcliste