| Assigned |
Due date | Read |
Exercises |
|---|---|---|---|
| 1/19 |
1/24 |
p. xiii pp. 1-10 |
Turn in on 1/24: 1.1 (on p. 19) |
| 1/24 |
1/31 |
pp. 11-14 |
1.2-5 Turn in on 1/26: Show that Rn over R with the usual addition and scalar multiplication is a vector space. |
| 1/26 |
1/31 |
pp. 11-14 |
Turn in on 1/31: Let F=R
and V={f: R->R}=the
set of functions from the reals to the reals. Define addition and on V
by (f+g)(x)=f(x)+g(x). For c in F and f in V, define scalar
multiplication by (cf)(x)=xf(x) (where c is a scalar). That is if f, g:R->R then f+g:R->R is the function whose value at the
real number x is f(x)=g(x). Similarly, if c is a real number and f:R->R then cf:R->R is the function whose value at the
real number x is cf(x). Verify that V is a vector space over F. |
| 1/31 |
2/7 |
pp. 12-16 |
1.5-9 Turn in on 2/7: Redo the exercise that was originally due on 1/31. |
| 2/7 |
2/14 |
pp. 14-18 |
1.10-15 Turn in on 2/9: Let V be a vector space and W a nonempty subset of V. Prove that W is a subspace of V if and only if av+bw is in W for all a,b in F and all v,w in W. |
| 2/14 |
2/21 |
pp. 21-25 |
Homework holiday. Prepare for the upcoming exam on 2/16. |
| 2/21 |
2/28 |
pp. 25-30 |
2.1-3, 5, 6 (Hint: If you have a list (f1, ..., fn),
pick n+1 points x1, ..., xn+1in [0,1] evaluate
the functions at those points to get n vectors v1, ..., vn
in Rn+1, find a vector w=(y1, ..., yn+1)
in Rn+1 which is not in the span{v1, ..., vn},
construct a polynomial p whose values at x1, ..., xn+1
are y1, ..., yn+1, and prove p is not in span{f1,
..., fn}.) Turn in on 2/23: Let V be a vector space and v1, ..., vn vectors in V. Prove that span{v1, ..., vn} is a subspace of V. |
| 2/28 |
3/7 |
pp. 31-34 |
2.8-12 Turn in on 3/7: 2.15 |
| 3/7 |
3/14 |
pp. 37-44 |
2.14, 16, 17 3.1, 2 Turn in on 3/9: Let F be some field, and V be the vector spaces of sequences over F (see p. 10 in yout textbook). Prove that the right shift T(x1, x2, ...) = (0, x1, x2, ...) and left shift U(x1, x2, ...) = (x2, x3, ...) are linear maps from V to V. Turn in on 3/14: 3.1 |
| 3/14 |
3/21 |
pp. 45-53 |
Homework holiday. Prepare for the upcoming exam on 3/21. |
| 3/21 |
4/11 |
pp. 53-58 |
3.3, 5-7, 11, 18, 21 Turn in on 3/9: Let F be some field, and V be the vector spaces of sequences over F (see p. 10 in yout textbook). Prove that the right shift T(x1, x2, ...) = (0, x1, x2, ...) and left shift U(x1, x2, ...) = (x2, x3, ...) are linear maps from V to V. Turn in on 3/14: 3.1 |
| 3/23 |
4/18 |
pp. 56-58, Ch. 4 |
3.14, 15 4.2-5 Turn in on 4/13: 3.22 Turn in on 4/18: 5.2 |
| 4/18 |
4/25 |
pp. 75-84 |
5.3-5, 8, 9, 11 Turn in on 4/25: 5.3 |
| 4/25 |
5/2 |
pp. 85-90 |
5.2, 7, 10, 12 (They mean every nonzero vector in V is an
eigenvector.), 13 (Hint: Use exercise 2 to prove that every
1-dimensional subspace is invariant under T, then use exercise 12), 17 Turn in on 5/2: 5.12 |
| 5/2 |
5/9 |
pp. 91-93 |
5.2.14, 15, 18, 20, 21 |