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Problem 191

Let \(U\) and \(W\) be the following subspaces of \(R^{4}\). $$ \begin{aligned} &\mathrm{U}=\\{(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}): \mathrm{b}+\mathrm{c}+\mathrm{d}=0\\} \\ &\mathrm{W}=\\{(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}): \mathrm{a}+\mathrm{b}=0, \mathrm{c}=2 \mathrm{~d}\\} \end{aligned} $$ Find the dimension and a basis of: (i) \(\mathrm{U}_{2}\) (ii) \(\mathrm{W}\), (iii) \(\mathrm{U} \cap \mathrm{W}\).

Expert verified

In summary, for the given subspaces U and W of \(R^4\), we have:
(i) The dimension of U_2 is 3, and its basis is \(\{(1,0,0,0),(0,1,-1,0),(0,0,-1,1)\}\).
(ii) The dimension of W is 2, and its basis is \(\{(-1,1,0,0),(0,0,2,1)\}\).
(iii) The dimension of U ∩ W is 2, and its basis is \(\{(-1,1,0,0),(0,0,2,1)\}\).

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