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2.6 Dual Spaces 1

# 2.6 Dual Spaces 1 - 2.6 2.6 Dual Spaces 1(a F(linear...

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2.6. 2.6. Dual Spaces 1. (a) F (linear transformation from V into its field of scalars F is called a linear func- tional) (b) T ( f : F F, [ f ] Mat 1 × 1 ( F )) (c) T dimV * =dim( L ( V, F ))=dim V dim F =dim V V V * (d) T For a vector space V , we can define the dual space of V i.e. ( L ( V, F )) = V * Then V is the dual space of V * (( V * ) * = V ) Every vector space is the dual of some vector space (e) (example) V = R 2 , F = R β = { e 1 = 1 0 , e 2 = 0 1 } : So V * = L ( V, R ) : 1 × 2 matrices and e * 1 = (1 , 0) , e * 2 = (0 , 1) i.e. β * = { e * 1 = (1 , 0) , e * 2 = (0 , 1) } Now if we define T : V V * by T ( e 1 ) = (1 , 1) , T ( e 2 ) = (1 , - 1) Since { T ( e 1 ) , T ( e 2 ) } = T ( β ) a basis of V * , then clearly T is an isomorphism But T ( β ) = { T ( e 1 ) = (1 , 1) 6 = e * 1 , T ( e 2 ) = (1 , - 1) 6 = e * 2 } 6 = β * 123 PNU-MATH

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2.6. (example 2) V = R , F = R , f : V F i.e.V * = F and β * = { 1 * } ( β = { 1 } ) But T : R R is an isomorphism a 7→ 2 a T ( β ) = { T (1) } = { 2 id } 6 = β * = { id } (f) T T : V W, T t : W * V * by T t ( g ) = gT ( T t ) t : ( V * ) * ( W * ) * (g) T V W T : V W : an isomorphism ⇔ ∃ [ T ] γ β : invertible ([ T ] γ β ) t = ([ T t ] γ * β * ) : invertible T t : W * V * : an isomorphism V * W * (h) F f : D n ( R ) R by f ( g ( x )) = g 0 ( x ) , g ( x ) = D n ( R ) but in case g ( x ) = x 2 , f ( g ( x )) = g 0 ( x ) = 2 x is not in R It’s not a linear functional 2.
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