Addition of linear subspaces
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Exercise:
Let mathcalUWsubset V be two finite dimensional linear subspaces of a vector space V. The following are equivalent: abcliste abc dimmathcalU+W dim mathcalU+dim mathcalW abc dimmathcalUcapmathcalW abc mathcalUcapmathcalW abc Every vin mathcalU+mathcalW can be written in a unique way as vu+w with uin mathcalU winmathcalW. abc If u+w where uinmathcalUwinmathcalW then uw. abcliste
Solution:
Proof of aiff b follows from a previous Proposition Proof of biff c because there is only one linear subspace of dim the space . Proof of cLongrightarrow d By definition every vinmathcalU+W can be written as vu+w. To see the uniqueness suppose that vu+wu'+w' where uu'in mathcalU ww'in mathcalW Longrightarrow u-u'w-w'in mathcalUcapmathcalW Longrightarrow uu' and w'w. Proof of dLongrightarrow e + so if u+w and d holds then uw. Proof of eLongrightarrow c Let vin mathcalUcapmathcalW. We have v+-v. If e holds we have v.
Let mathcalUWsubset V be two finite dimensional linear subspaces of a vector space V. The following are equivalent: abcliste abc dimmathcalU+W dim mathcalU+dim mathcalW abc dimmathcalUcapmathcalW abc mathcalUcapmathcalW abc Every vin mathcalU+mathcalW can be written in a unique way as vu+w with uin mathcalU winmathcalW. abc If u+w where uinmathcalUwinmathcalW then uw. abcliste
Solution:
Proof of aiff b follows from a previous Proposition Proof of biff c because there is only one linear subspace of dim the space . Proof of cLongrightarrow d By definition every vinmathcalU+W can be written as vu+w. To see the uniqueness suppose that vu+wu'+w' where uu'in mathcalU ww'in mathcalW Longrightarrow u-u'w-w'in mathcalUcapmathcalW Longrightarrow uu' and w'w. Proof of dLongrightarrow e + so if u+w and d holds then uw. Proof of eLongrightarrow c Let vin mathcalUcapmathcalW. We have v+-v. If e holds we have v.