- #1

MastersMath12

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a) T = {(x1, x2, x3) | x1x2x3 = 0}

b) T = {(x1, x2, x3) | x1 - x3 = 0}

c) T = {(x1, x2, x3) | x1 = 0}

d) T = {(x1, x2, x3) | x1 = 1}

2) Let U be a k-vector space, where k is any field. Let V be a subspace of U. Assume that dimk V = dimk U. Show that U = V

3) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Za.

4) Let T: U--->V be a linear map. Show that T(0u) = 0v

5) Find an example of vector space V and two subspaces W ⊂ V and Z ⊂ V such that Z ∪ W is not a subspace?

6) Show that T = {a + b√3 | a,b∈T } is a field (a subfield of R). Show that M = {a + b√3 | a,b∈ L} is not a field