Problem 9: Let (V, (-,–)) be a n-dimensional real inner product space. For any ve V, denote by ov the corresponding element in VV, i.e., ov(w) = (v, w) for every w € V. (a) Show that two nonzero vectors v1 and v2 in V are colinear if and only if ker(yv,) = ker(pva). (b) Show that more generally, given a set of vectors S = {v1,·… , v,} in V, S is linearly independent if and only if the subspace ker(øv,) n..nker(øv,) has dimension exactly n –r. (Hint: If you cannot come up with a direct argument, try to induct on r.)

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Q: 3c

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