Page 102
In 3,5,7 the given set and operations are not a vector
space. Cross out the vector-space property that fails.
3. V = {x: x>0}, xy = x+y, cx = cx.
1. u
v = v
u 5. c
(u
v) = c
u
c
v
2. u
(v
w) = (u
v)
w 6. (c+d)
u = c
u
d
u
3. For some 0, u
0 = u 7. c
(d
u) = (cd)
u
4. u
-u = 0 for some -u 8. 1
u = u
5. V = {(x,y,z): x,y,z are real numbers}, (x, y,z)(r, s, t)
= (x+r, y+s, z+t), c(r, s, t)= (r, 1, t). .
1. u
v = v
u 5. c
(u
v) = c
u
c
v
2. u
(v
w) = (u
v)
w 6. (c+d)
u = c
u
d
u
3. For some 0, u
0 = u 7. c
(d
u) = (cd)
u
4. u
-u = 0 for some -u 8. 1
u = u
7. V = {(x,y): x,y are real numbers},
(x, y)(r, s) = (x+r, y+s), c(r, s)= (0, 0).
1. u
v = v
u 5. c
(u
v) = c
u
c
v
2. u
(v
w) = (u
v)
w 6. (c+d)
u = c
u
d
u
3. For some 0, u
0 = u 7. c
(d
u) = (cd)
u
4. u
-u = 0 for some -u 8. 1
u = u
9. Prove that the following set and operations form a
vector space. Answer is at the bottom of each problem.
V = {real-valued continuous functions}, fg = f+g where
(f+g)(t) = f(t)+g(t), rf = rf where (rf)(t) = r(f(t)).
(1) Prove fg = gf.
(fg)(t) = f(t)+g(t) = g(t)+f(t) = (gf)(t)
(2) Prove f(gw) = (fg)w.
(f(gw))(t) = f(t)+(gw)(t) = f(t)+(g(t)+w(t))
=(f(t)+g(t))+w(t) = (fg)(t)+w(t) = ((fg)w)(t)
(3) What serves as the zero vector 0 ? ______ .
Prove f0 = f.
(f0)(t) = f(t)+0(t) = f(t)+0 = f(t)
(4) Given f what is the negative (as a vector) -f? _______.
Prove f-f = 0.
(f-f)(t) = f(t)+-f(t) = 0 = 0(t)
(5) Prove c(fg) = cf cg.
(c(fg))(t) = c((fg)(t)) = c(f(t)+g(t))
= cf(t)+cg(t)=(cf)(t)+(cg)(t)=(cf cg)cf cg
(6) Prove (c+d)f = cf
df.
((c+d)f)(t) = (c+d)(f(t) = cf(t)+d(t) =
(cf)(t) +(df)(t) = (cf
df)(t)
(7) Prove c(df) = (cd)f.
(c(df) )(t) = c(df)(t) = c(df(t)) = cd(f(t)) = (cdf)(t)
(8) Prove 1f = f.
(1f)(t) = 1(f(t)) = f(t)
In 12, 14, cross out the vector-space properties that fail.
13. V = (-, ) = all reals
uv = uv, cu = c+u.
1. u
v = v
u 5. c
(u
v) = c
u
c
v
2. u
(v
w) = (u
v)
w 6. (c+d)
u = c
u
d
u
3. For some 0, u
0 = u 7. c
(d
u) = (cd)
u
4. u
-u = 0 for some -u 8. 1
u = u
Answers
3. Properties 3, 4, 5, 6, 7.
5. Properties 5, 6, 8.
7. Property 8.
13. Properties 4, 5, 6, 7, 8.
Math 311 Hw 9 Recommended problems, don't turn this in.
Exam 1 Wed. This is due Friday. Hw 102: 4,6, 8, 10, 12, 14. Recommended 102: 3, 5, 7, 9, 13. Answers 2.2, 539.