1.I .5E. 5 \mathrm{E} \quad

Linear Mathematics | Part IB, 2003

Let VV be the subset of R5\mathbb{R}^{5} consisting of all quintuples (a1,a2,a3,a4,a5)\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right) such that



a1+2a2+3a3+4a4+5a5=0a_{1}+2 a_{2}+3 a_{3}+4 a_{4}+5 a_{5}=0

Prove that VV is a subspace of R5\mathbb{R}^{5}. Solve the above equations for a1a_{1} and a2a_{2} in terms of a3,a4a_{3}, a_{4} and a5a_{5}. Hence, exhibit a basis for VV, explaining carefully why the vectors you give form a basis.

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