Download problem set #14

Problem Set #14
MATH 110 : 2015–16
Due: Friday, 15 January 2016
1. (a) Determine whether R2 , with the usual scalar multiplication but addition defined by
” ı ” ı ”
ı
v1
w1
v1 ` w1 ` 1
for all v1 , v2 , w1 , w2 P R,
v2 ` w2 :“ v2 ` w2 ` 1
is a real vector space. If it is not, list all of the axioms that fail to hold.
(b) Given the complex number c “ a ` bi where a, b P R, the conjugate is defined by c :“ a ´ bi.
Determine whether C2 , with the usual vector addition but scalar multiplication defined by
” ı ” ı
z
cz
for all c, z1 , z2 P C,
c z1 :“ c z1
2
2
is a complex vector space. If it is not, list all of the axioms that fail to hold.
2. Let V be a K-vector space. Prove that the intersection of any collection of linear subspaces of V
is also a linear subspace.
3. Given functions f1 , f2 , . . . , fn in Cn´1 pRq, the real vector space of all function with continuous
pn ´ 1q-th derivatives, the determinant
»
fi
f1 pxq
f2 pxq
¨¨¨
fn pxq
— f11 pxq
f21 pxq
¨¨¨
fn1 pxq ffi
—
ffi
— 2
f22 pxq
¨¨¨
fn2 pxq ffi
W pxq :“ det — f1 pxq
ffi
—
ffi
..
..
..
.
.
–
fl
.
.
.
.
pn´1q
f1
pn´1q
pxq f2
pxq ¨ ¨ ¨
pn´1q
fn
pxq
is called the Wronskian. If the Wronskian is nonzero at some point x P R, then show that the
functions f1 , f2 , . . . , fn are linearly independent.
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