1. LAHW#12 Due December 13, 2010

2. 5.2 Bases and Dimension 42. –Criticize this argument: We have three vectors u 1 = (1, 3, 2), u 2 = (-2, 4, 5), and u 3 = (-1, 7, 7). We notice that u 3 is a linear combination of u 1 and u 2. Furthermore, u 2 is a linear combination of u 1 and u 3. Using Theorem 8 twice, we conclude that both u 2 and u 3 can be removed from the set { u 1, u 2, u 3 } without affecting the span of that set.

3. 5.2 Bases and Dimension 51. –Substantiate that if a vector space V contains a linearly independent set of k vectors and if another set of k vectors spans V, then Dim( V )= k.

4. 5.2 Bases and Dimension 66. –Let S be a subset of a vector space, V. Suppose that S spans V, and that there is at least one vector in V that has a unique expression as a linear combination of the vectors in S. Is S necessarily a basis for V.

5. 5.2 Bases and Dimension 81. –Give an example of a matrix in reduced row echelon form for which the co-domain, domain, range, and kernel have dimensions 8, 7, 4, and 3, respectively.

6. 5.3 Coordinate Systems 6. –Define these six vectors: u 1 = (1, 4, 7, 2), u 2 = (3, 1, 1, 5), u 3 = (-3, 10, 19, -4), v 1 = (7, 1, 6, 5), v 2 = (-2, 1, 3, 1), and v 3 = (11, 3, 2, 9). Find a linear transformation L such that L(u i ) = v i, for 1 ≦ i ≦ 3.

7. 5.3 Coordinate Systems 12. –Let T : V → W, where V has basis B = {b 1, b 2 }, W has basis C = {c 1, c 2, c 3 }, T(b 1 ) = 3c 1 + 5c 2 – 7c 3, and T(b 2 ) = 2c 1 – c 2 + 4c 3. Find M so that [T(x)] C = M[x] B. If [x] B = [7, -2] T, then what is [T(x)] C ?

8. 5.3 Coordinate Systems 21. –Give an argument for this assertion. If B is a basis for some vector space and B = {u 1, u 2, …, u n }, then for each i, the coordinate vector for u i is e i (the i th standard unit vector in R n ). Thus, in symbols, [u i ] B = e i.

9. 5.3 Coordinate Systems 27. –Explain that if one of A and B is invertible, then AB BA. Is the invertibility hypothesis necessary? (An example or theorem is needed.)