Academy of Chinese Culture & Health Sciences Linear Algebra Questions

Linear algebra Questions

1. Heffron’s text: Exercises 1.18 and 1.19, (pg. 87).
2. Let Pn be polynomials of degree n or less in x.
a. Show that P2 is a subspace of P3.
b. Let U be the subset of Pn comprising all polynomials such that p(1) + p'(1) = 0 and p(2) = 0. Is U a
subspace? What happens if the condition is changed to p(1) + p'(1) = 0 or p(2) = 0?
=
=
3. (i) Consider U = {f € C[a, b] | Sá f(x)dx = 0}. Determine whether U is a subspace of C[a, b].
(ii) Consider U = {x = (x1 x2)7 € R2|21X2 = 0}. Determine whether U is a subspace of R2.
4. Heffron’s text: Exercises 2.20(c,d), 2.22(b, c), 2.26(b, d), (pgs. 97 – 98).
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5. (i) Consider the sets S1 = {(1 1 2)7,(2 -1 1)T} and S2 = {(1
3)T}.
Show that span (S1)
= span (S2).
(ii) Determine whether or not the set of polynomials S = {1 – x, 2x + x2,1+x+x2} is linearly independent.
7:03 PM Sun Jun 7
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S.
b
f)(x) dx
f(x
(x) dx
b
$ (8 +91lade – $[70) 9) dx = $63
) dx + focus.de
Sf(x) g(x]
s(f+g)(x)dx = o to
$lebat
S(fig)(x) =
)
0
2
By definition of º, ftg E .
u tu
☆ for all fev, kf EU with K E IR.
let f be function of u. Let a be a constant. So.
a
2
b
$(kr)(x )dx = xfFcodex
Scx f(x) dx = 0
2
b
).