Question

Show if S{uyor, Up} spans H and is linearly dependent, then å vector can be removed from S and it will still span H.

Answer 1

Solne Given, s={U, U₂, U3, Vp? Spans H. and is dependent. linearly. ماک Proves if a vector is removed from and it will Still s. Span he Vectors aire a Va Up dependent it linearly & V, t t 2 + (3 Uz + - + Cp Up 0. Where are Some is the lp non-Zero. let required Vector removed from s .

Date Page -t & @ Since eqnili gives. Up = -cp-1 fc u + C V₂ + (z Uz + Cp.. Up-1} H= Span {U2, U3 Up} Then for Given any x in H. , we write may x= 1, U, + dzų t...t dp- Up-t + dp Up. for Suitable sealers. Lot de Substituting the values of Up from (2). in (3) nel have t or x = 1, V, + 2 Uq tap-1 Up-) + xp{-(p' (al + C Uqt top? x = (x - lp Conta) + (laap (p all + (xp-1-dp (p (p-1) Up-1 linear Compination of + x is q We see that Vectors ७। 1927 137 Up-t ང Since e is an arbitrary vectors of_H

Date Page { vis Vzy--- Up-r? Span H. Idence, proved it one of the vectors of is removed then remaining vectors still span H. Set S say up c