College Algebra Exam Review 166

# College Algebra Exam Review 166 - 176 3 PRODUCTS OF GROUPS...

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176 3. PRODUCTS OF GROUPS order to view the two variables on an equal footing, let us introduce a new notation for the pairing, f.v/ D h v;f i . This function of two variables is bilinear , that is, linear in each variable separately. This means that for all scalars ˛ and ˇ and all v;v 1 ;v 2 2 V and f;f 1 ;f 2 2 V ± , we have h ˛v 1 C ˇv 2 ;f i D ˛ h v 1 ;f i C ˇ h v 2 ;f i ; and h v;˛f 1 C ˇf 2 i D ˛ h v;f 1 i C ˇ h v;f 2 i : Linearity in the ﬁrst variable expresses the linearity of each f 2 V ± , h ˛v 1 C ˇv 2 ;f i D f.˛v 1 C ˇv 2 / D ˛f.v 1 / C ˇf.v 2 / D ˛ h v 1 ;f i C ˇ h v 2 ;f i : Linearity in the second variable, on the other hand, reﬂects the deﬁnition of the vector operations on V ± , h v;˛f 1 C ˇf 2 i D .˛f 1 C ˇf 2 /.v/ D ˛f 1 .v/ C ˇf 2 .v/ D ˛ h v;f 1 i C ˇ h v;f 2 i : The following observation applies to this situation: Lemma 3.4.3. Suppose that V and W are vector spaces over a ﬁeld K , and b W V ± W ²! K is a bilinear map. Then b induces linear maps ± W V ²! W ± and ² W W ²! V ± , deﬁned by
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