Problem:
Solution:
This is easy.
In exercise 3, we just shown $ \langle x + xy, y + xy, x^2, y^2 \rangle = \langle x, y \rangle $.
In exercise 4, we just shown if $ \langle f_1, \cdots, f_s \rangle = \langle g_1, \cdots, g_t \rangle $, then $ \mathbf{V}(f_1, \cdots, f_s) = \mathbf{V}(g_1, \cdots, g_t) $.
This is just applying the two facts we have just proved.
Solution:
This is easy.
In exercise 3, we just shown $ \langle x + xy, y + xy, x^2, y^2 \rangle = \langle x, y \rangle $.
In exercise 4, we just shown if $ \langle f_1, \cdots, f_s \rangle = \langle g_1, \cdots, g_t \rangle $, then $ \mathbf{V}(f_1, \cdots, f_s) = \mathbf{V}(g_1, \cdots, g_t) $.
This is just applying the two facts we have just proved.
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