Question

1. Why do S1 and S2 exist?

2. Where does equation 2 come from?

subsets of a vector space and let S, be a subset of S2. Then Let Si and S2 be finite subsets of a vector the following statements are true: (a) If S, is linearly dependent, so is S2. (b) If S2 is linearly independent, so is Si. Proof Let Si = {V1, V2, ..., vk and S2 = {V1, V2, ..., Vk, Vx+1, ..., Vm). We first prove (a). Since S, is linearly dependent, there exist an, 42, ..., ak zero, such that avi + a2V2 + ... + ax V = 0. Then (2) av + a2V2 + ... + ax V + OVX+1 + ... + Ovm = 0. Since not all the coefficients in (2) are zero, we conclude that S2 is linearly depen- dent. . 11m alant

Answer 1

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-0 a dovt. td VI takt vicept.. fLm Vis Since si is .D (mearly depond out). - 7 d., & K EF, not all zero. Sit divit. & de Vic - 5 of whore &. z o for some i kick then By O ase dito for some isl'ék So is Innoorly depondent

6 If Sa is too linearly independent then prove that s, is linedly independent it LV, t. tak k = 0 for some di EF. PV, V, Vic] =S, che Since Sa is t. I (lineady indepardent) = 2 = 0, 0,50 - 20 woledo s is L. I