College Algebra Exam Review 166

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online