Assume V is a subspace of Rn and dim(V) = k. Show any set T= which...

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Assume V is a subspace of Rn and dim(V) = k. Show any set T= <u1,u2,...uk > which is linearly independent is also a basis of V. Assume V is a subspace of Rn and dim(V) = k. Show any set T= <u1,u2,...uk > which is linearly independent is also a basis of V.

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Soli Let V be subspace of IR sit dim Vk and T 4 4 x EV od I is linearly independent Now Tis basis of Viff Tis linearly independent and it spans V As T is already given I already given to be lineasly independent So to show I is basis of V it is sufficient to show that I spansk Let UEV be arbitrary So AD T x is linearly independent and dim VK set Containing kri vectors must be linearly dependent ie 2 un is linearly dependent any sit a n uk tau 0 where not all 9 19 9 9 0 if then we have auit gut 0 a a au a Thus ato then from 0 we get o I is linently independen ... View the full answer