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Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 1 of 6 Chapter 5. General Vector Spaces. Section 5.1: Real Vector Spaces. Deﬁnition: Vector Space.. Let V be an arbitrary nonempty set of objects
on which two operations are deﬁned, addition and multiplication by scalars
(numbers). By addition we mean a rule for associating with each pair of
objects u and V in V an object u + V, called the sum of u and V; by scalar
multiplication, we mean a rule for associating with each scalar k and each
object u in V an object ku, called the scalar multiple of u by k. If the
following axioms are satisﬁed by all objects u, V, W in V and all scalars k
and l, then we call V a vector space and we call the objects in V vectors. @If u and V are objects in V, then u + V is in V. CLOSUVE 0N6“ MQT‘WJ 2. u——v=v+u. 3. u——(v+w)=(u+v)+w. 4. There is an object 0 in V called the zero vector for V, such that
0+u=u+0=u 5. For each u in V, there is an object —u in V, called the negative of 11,
such that u + (—u) = (—u) + u = 0. If k is any scalar and u is any object in V, then ku is in V. 095°“ WW9
SCAUW. nun'53.
7. k(u + V) 2 kn + kv.
8. (k + l)u 2 kn + Zn.
9. k(lu) = (kl)u. 10. 1n = u. Example: Euclidean nspace, R" with the usual addition and scalar mul
tiplication Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 2 of 6 Example: Vector Space of 2 X 2 matrices. Show that the set of all 2 X 2
matrices with real entries is a vector space if vector addition is deﬁned to
be matrix addition and vector scalar multiplication is deﬁned to be matrix
scalar multiplication. To show this, we have to show that each of the vector space axioms (from the previous page) hold for any three 2 X 2 matrices u, V, W, and any scalars
k, l. “ (5 \‘A‘ 09. ‘$ELO)J6$ To“
1. If u an V are objects in V, then u + V is in V. 5»! (AK! \l= M17.  ‘5‘?“ “ml .97?“ d“ “9“ T“ u¢$“&"~"\‘
Uvn um. . Unﬁt V“ unﬁ‘ U“. \]
1H6 )3 u ‘k \l = 6 .
a U1“? “1" “It 2.u+v=v+u. So u)in 58.3160 AgAgovE.
UJ! ‘— U‘“*V" u\1*u“\ = V\\“'uu Vt1*u\1\ ._ \l'l'g
Wu" Vu (L113 U11— sz+u11 V1.1} u?!
' “on “on.
3.u+(v+w)=(u+v)+w. worm 52¢) AS «WE 2. vg=[ ] Vll+wl V“. + w\1.‘\ [ U.“ +( Vu" Mu) “1,4. (“\1+w\1) ‘\ u+(\m~ '— EH
" ' ‘3 Vu’m’u Uri"wit ulna—(013M120 “12*(011ﬁwul 4. There is an object 0 in V called the zero vector for V, such that
0 + u = u + 0 = u. 0=[°° av. ~ 00 5. For each u in V, there is an object —u in V, called the negative of u,
such that u + (—u) = (—u) + u = 0. ‘11“ " u\‘l.
"I—L '=
“uh ' “21 Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 3 of 6 6. If k is any scalar and u is any object in V, then ku is in V. “‘La
Lg‘xku u kul" ‘1' u), 7. k(u+v)=ku+kv. wém L5 6.2 As Aeouz ’ \g Aw sum. U.“ + V“ uVL" “\t‘k  [ k(ua“+uu) ﬁlt“ ‘1‘?) a u“ ‘\’ k J“ ‘4 “\L* kv‘t 1—
I. kCu1\*U‘L\) \L(_q'nkuib) k U~u+ ‘4VH \L “11* kul \. (g+2\= \4[ (114* Uh uh.“ V11. 2.”: iétilLA; .
8. (k+l)u=ku+lu. you Fhﬁsy‘ «H15 oFﬁ. 9. k(lu) = (kl)u. 10. 1n = u. Dr. Doom Lin Alg MATH1850/2050 Section 5.1 Page 4 of 6 Example: Vector Space of Functions. Let V be the set of realvalued
functions deﬁned on the entire real line (—00, 00). If f = f(a:) and g 2 9(33)
are two such functions and k is any real number, deﬁne the sum function
f + g and the scalar multiple kf, respectively, by (f + g)($) = f (56) + 9(36) and (WW) = left?) In other words, the value of the function f + g at a: is obtained by adding
together the values of f and g at 3:. Similarly, the value of kf at a: is k
times the value of f at as. This is the usual addition and multiplication by a
constant, which you are familiar with. If f and g are vectors in V, then to
say that f = g is equivalent to saying that f 2 9(33) for all as. The vector
space V is often labeled as F(—oo, 00). Show that V is indeed a vector space. cLosuue aﬂoat. Aoéufon a \r— Q 8.36 ?(o°,o°) ‘Iueo {+13 4cm»).
(501 Foe Ayw x6 It J (Cu) 8. 36:) ALE OEFquIsJ umqu. ’
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= Cw(*b)(x) r\' ka)
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= M 3. \«QM. ‘lcu FI'UCSH Tue 2,651: Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 5 of 6 (A) Example: 6 page 225 from text. Let V be any plane through the origin in
R3. This example shows that the points in V form a vector space under the
standard addition and scalar multiplication operations for vectors in R3. @ Example: Let V be the line y = a: + 1 in R2. Deﬁne addition and scalar
multiplication as the standard addition and scalar multiplication of vectors
in R2. Show that the points in V do not form a vector space. F“ G) We“ 0”“ “EW @ 4® (WE OWE“ % Mons A¢E GoaGina) mm
W} (SEm'c A vacrm. some). SJ” g1cu‘au1Nﬁ) 2‘ 2=(\’\;\’1.V~s) Am 1» #1 “AH:
qx \lo‘3 *0; =0
so Au\+bu1CU‘3=O 5‘ QVVEEV'L‘LCUbao. pm) 95+! —. (“ﬂ—u“ ,utwa, ,UksNs) & 0~(U~.W\)+S(u;w,,)4 C(u..,w«..\= ans‘9 Lit\Cuz, + qvﬂ‘odﬁcv3
= O ‘\' O '3 0.
So (45w 6 @ 8A1 u A5 meme 8. l; Any <(ALA9. Ckéll)
\éc; : (V‘M )\4“1)\‘U‘5) aﬁub+$<kua+cﬂcuﬁ= \L (Qu.+lOU~LL Cu!) = L‘O " O .
so lug. G so \I (s A vamp, $0Acc. ' ‘ \1
F09. @ : Cuecv. @ 1F 15 ZgéU (on LNG B‘XJM), mace. u: (£4! e 30 u=(u‘,u¢) ’ um: LU“ A ytcv)u1\w\1\\' VLlVl'Vl .— uﬁul: Cu‘+l)r(\1v\\)= (\L‘+V\\‘\'2. f (u.\\\J\\\'\
5., guy C—\} SO x] is Not A decwt sax€ ’— Dr. Doom Lin Alg MATHl850/2050 Section 5.1 Page 6 of 6 The Zero Vector Space. Let V consist of a single object, which we Will
denote by 0, and deﬁne addition as 0+0=0 and scalar multiplication as
k0 = 0 for all scalars k. It is easy to check that all the vector space axioms are
satisﬁed. We call this the zero vector space. Theorem 5.1.1: Properties of Vector Spaces. Let V be a vector space,
11 be a vector in V, and k a scalar; then: (a) 0n = 0.
(b) k0 = 0.
(c) (—1)u = —u. (d) Ifku=0,thenk=00ru=0. @ 0 g [email protected])g = Qg 1:02. émce OgéU J l1's ween(us EXIST: Ot3 $9 Og+C—Og):OgLOg+C'Og)
So Q=Og+Q , Hence avg ® \LQ .. \c(O_iQ).—_ \ggrkg AM ~—lLQ To earn sides, a(,ET lLQ=Q. Q GI)“.\.g—;(—\)B.\. s (—lEog =0té=g Agaat E*(_\)%=g ASWELL (1) 3w barQ JUL#0 Cum“ SawQ) $0 téll ...
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This note was uploaded on 11/10/2009 for the course MATH 1850 taught by Professor Mihaibeligan during the Spring '09 term at UOIT.
 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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