6. Let e = (v1, ..., vn) be an ordered basis of an n-dimensional vector space V , and let Le : V → K n be the coordinate map given by

Le(v) = (a1, a2, ..., an) ∈ K n , where the ai are such that v = a1v1+a2v2+· · ·+anvn. Show that Le is a linear isomorphism.

3. Let f : R 2 → R 2 be the function given by f(x, y) = (x, 0). (a) Is f one-to-one? (b) Is f onto? (c) What is im(f)? (d) What is f −1 (1, 0)?

5. Let L : R 2 → R 2 be given by L(x, y) = (x + y, x − y), and let M : R 2 → R be given by M(x, y) = 2x − y. (a) Determine L −1 (0, 0) and M−1 (0). (b) Determine a formula for (M ◦ L)(x, y).

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