vector

1. VECTOR SPACES
4.1 VECTORS IN RN
4.2 VECTOR SPACES
4.3 SUBSPACES OF VECTOR SPACES
4.4 SPANNING SETS AND LINEAR INDEPENDENCE
4.5 BASIS AND DIMENSION
4.6 RANK OF A MATRIX AND SYSTEMS OF LINEAR
EQUATIONS
4.7 COORDINATES AND CHANGE OF BASIS

2. WEL COME
• PATEL SUJAY A (130390119086)
• PATEL TRUPAL S (130390119087)
• PATEL KUNTAK (130390119070)
• PATEL PARTH (130390119081)
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3. 4.1 VECTORS IN RN
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 An ordered n-tuple:
a sequence of n real number ( , , , ) 1 2 n x x  x
 n-space: Rn
the set of all ordered n-tuple

4. 4
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n = 4
R4 = 4-space
= set of all ordered quadruple of real numbers
(x1, x2 , x3, x4 )
R1 = 1-space
= set of all real number
n = 1
n = 2 R2 = 2-space
= set of all ordered pair of real numbers (x1, x2 )
n = 3 R3 = 3-space
= set of all ordered triple of real numbers (x1, x2 , x3 )
 Ex:

5. • Notes:
(1) An n-tuple can be viewed as a point in Rn
with the xi’s as its coordinates.
(2) An n-tuple can be viewed as a vector
in Rn with the xi’s as its components.
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 Ex:
a point
( x1, x2 )
a vector
( ) x1, x2
(0,0)
( , , , ) 1 2 n x x  x
( , , , ) 1 2 n x x  x
( , , , ) 1 2 n x = x x  x

6. ( n ) ( n ) u ,u , ,u , v ,v , ,v u = 1 2  v = 1 2  (two vectors in Rn)
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 Equal:
u = v if and only if n n u = v , u = v , , u = v 1 1 2 2 
 Vector addition (the sum of u and v):
( ) n n + = u + v , u + v , , u + v u v 1 1 2 2 
 Scalar multiplication (the scalar multiple of u by c):
( ) n c cu ,cu , ,cu u = 1 2 
 Notes:
The sum of two vectors and the scalar multiple of a vector
in Rn are called the standard operations in Rn.

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 Negative:
( , , ,..., ) 1 2 3 n - u = -u -u -u -u
 Difference:
( , , ,..., ) 1 1 2 2 3 3 n n u - v = u - v u - v u - v u - v
 Zero vector:
0 = (0, 0, ..., 0)
 Notes:
(1) The zero vector 0 in Rn is called the additive identity in Rn.
(2) The vector –v is called the additive inverse of v.

8. • Thm 4.2: (Properties of vector addition and scalar multiplication)
Let u, v, and w be vectors in Rn , and let c and d be scalars.
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(1) u+v is a vector in Rn
(2) u+v = v+u
(3) (u+v)+w = u+(v+w)
(4) u+0 = u
(5) u+(–u) = 0
(6) cu is a vector in Rn
(7) c(u+v) = cu+cv
(8) (c+d)u = cu+du
(9) c(du) = (cd)u
(10) 1(u) = u

9. • Ex 5: (Vector operations in R4)
Let u=(2, – 1, 5, 0), v=(4, 3, 1, – 1), and w=(– 6, 2, 0, 3) be
vectors in R4. Solve x for x in each of the following.
(a) x = 2u – (v + 3w)
(b) 3(x+w) = 2u – v+x
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Sol: (a)
x u v w
= - +
u v w
2 ( 3 )
= - -
2 3
= - - - - -
(4, 2, 10, 0) (4, 3, 1, 1) ( 18, 6, 0, 9)
= - + - - - - - + -
(4 4 18, 2 3 6, 10 1 0, 0 1 9)
= - -
(18, 11, 9, 8).

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(b)
x + w = u - v +
x
3( ) 2
x + w = u - v +
x
3 3 2
x - x = u - v -
w
3 2 3
x = u - v -
w
2 2 3
= - 1
-
( ) ( ) ( )
(9, , , 4)
= + - + -
2,1,5,0 2, , , 9, 3,0,
9
2
11
2
9
2
1
2
1
2
3
2
2 3
2
= -
-
- - -
x u v w

11. • Thm 4.3: (Properties of additive identity and additive inverse)
Let v be a vector in Rn and c be a scalar. Then the following is true.
(1) The additive identity is unique. That is, if u+v=v, then u = 0
(2) The additive inverse of v is unique. That is, if v+u=0, then u = –v
(3) 0v=0
(4) c0=0
(5) If cv=0, then c=0 or v=0
(6) –(– v) = v
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12. The vector x is called a linear combination of ,
if it can be expressed in the form
• Linear combination:
, , , : scalar 1 2 n c c  c
12/107
 Ex 6:
Given x = (– 1, – 2, – 2), u = (0,1,4), v = (– 1,1,2), and
w = (3,1,2) in R3, find a, b, and c such that x = au+bv+cw.
Sol:
b c
- + = -
3 1
2
a b c
+ + = -
a b c
+ + = -
4 2 2 2
Þa =1, b = -2, c = -1
Thus x = u - 2v -w
n v , v ,..., v 1 2
n n x c v c v c v 1 2 = 1 + 2 ++

13.  Notes:
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A vector ( , , , ) in can be viewed as: 1 2 n u = u u  u Rn
[ , , , ] 1 2 n u = u u  u
ù
ú ú ú ú
û
u
é
=
ê ê ê ê
1
u
n u
ë

2
u
or
a 1×n row matrix (row vector):
a n×1 column matrix (column vector):
(The matrix operations of addition and scalar multiplication
give the same results as the corresponding vector operations)

14. Vector addition Scalar multiplication
u v  
u u u v v v
= + + +
( , , , ) ( , , , )
n n
u v  
u u u v v v
= + + +
[ , , , ] [ , , , ]
n n
ù
ú ú ú ú
û
u v u v u v
u v u v u v
é
=
ê ê ê ê
ë
ù
ú ú ú ú
û
1 2 1 2
u
1 2 1 2
é
=
ê ê ê ê
1
ù
u
ë
cu
1
cu
n n cu
u
u +
v
c c
 
2
2
u
u =
c c u u u
( , , , )
u =
c c u u u
[ , , , ]
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( , , , )
1 1 2 2
n n
+ = +

[ , , , ]
1 1 2 2
n n
+ = +

ú ú ú ú
û
é
ê ê ê ê
1 1
u v
1
1
  
ë
+
+
=
ù
ú ú ú ú
û
é
+
ê ê ê ê
ë
ù
ú ú ú ú
û
u
é
ê ê ê ê
u
ë
+ =
v
v
v
u v
n n n n u
2 2
2
2
u v
1 2

( , , , )
1 2
n
n
cu cu 
cu
=
1 2

[ , , , ]
1 2
n
n
cu cu 
cu
=

15. 4.2 VECTOR SPACES
• Vector spaces:
15/107
Let V be a set on which two operations (vector addition and
scalar multiplication) are defined. If the following axioms are
satisfied for every u, v, and w in V and every scalar (real
number) c and d, then V is called a vector space.
Addition:
(1) u+v is in V
(2) u+v=v+u
(3) u+(v+w)=(u+v)+w
(4) V has a zero vector 0 such that for every u in V, u+0=u
(5) For every u in V, there is a vector in V denoted by –u
such that u+(–u)=0

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Scalar multiplication:
(6) c u is in V.
(7) c ( u + v ) = c u + c v
(8) (c + d)u = cu + du
(9) c(du) = (cd)u
(10) 1(u) = u

17. • Notes:
17/107
(1) A vector space consists of four entities:
a set of vectors, a set of scalars, and two operations
V：nonempty set
c：scalar
( , ) ：
u u
+ = + vector addition
u v u v
· c = c
( , ) ：
scalar multiplication
(V, +, ·) is called a vector space
(2) V = {0}： zero vector space

18. • Examples of vector spaces:
( , , , ) ( , , , ) ( , , , ) 1 2 n 1 2 n 1 1 2 2 n n u u  u + v v  v = u + v u + v  u + v
vector addition
scalar multiplication
18/107
(1) n-tuple space: Rn
( , , , ) ( , , , ) 1 2 n 1 2 n k u u  u = ku ku  ku
(2) Matrix space: m (tnhe set of all m×n matrices with real values) V M ´ =
Ex: ：(m = n = 2)
ù
úû
é
êë
u + v u +
v
11 11 12 12
+ +
ù
= úû
é
+ úû ù
êë
é
êë
21 21 22 22
v v
11 12
21 22
u u
11 12
21 22
u v u v
v v
u u
ù
úû
é
= úû
êë
ù
é
u u
k
êë
ku ku
11 12
21 22
11 12
21 22
ku ku
u u
vector addition
scalar multiplication

19. 19/107
(3) n-th degree polynomial space:
V P (x) n =
(the set of all real polynomials of degree n or less)
n
n n p(x) q(x) (a b ) (a b )x (a b )x 0 0 1 1 + = + + + ++ +
n
nkp x = ka0 + ka1x ++ ka x ( )
(4) Function space: V = c(-¥(the ,¥set )
of all real-valued
continuous functions defined on the entire real
line.) ( f + g)(x) = f (x) + g(x)
(kf )(x) = kf (x)

20. 20/107
 Thm 4.4: (Properties of scalar multiplication)
Let v be any element of a vector space V, and let c be any
scalar. Then the following properties are true.
v =
0
(1) 0
0 =
0
v 0 v 0
c c
= = =
(2)
c
(3) If , then 0 or
v v
- = -
(4) ( 1)

21. • Notes: To show that a set is not a vector space, you need
only find one axiom that is not satisfied.
 Ex 6: The set of all integer is not a vector space.
ÎV , ÎR 2
1 1
(1 )(1) = 1
ÏV (it is not closed under scalar multiplication)
2
2
­ ­ ­ scalar
noninteger
 Ex 7: The set of all second-degree polynomials is not a vector space.
21/107
Pf: Let p ( x ) = x 2 and q(x) = -x2 + x +1
Þ p(x) + q(x) = x +1ÏV
(it is not closed under vector addition)
Pf:
integer

22. • Ex 8:
( , ) ( , ) ( , ) 1 2 1 2 1 1 2 2 u u + v v = u + v u + v
22/107
V=R2=the set of all ordered pairs of real numbers
vector addition:
scalar multiplication: ( , ) ( ,0) 1 2 1 c u u = cu
Verify V is not a vector space.
Sol:
1(1, 1) = (1, 0) ¹ (1, 1)
the set (together with the two given operations) is
not a vector space

23. 4.3 SUBSPACES OF VECTOR SPACES
• Subspace:
(V,+,·)
W f
þ ý ü
¹
W Í
V
23/107
: a vector space
: a nonempty subset
(W,+,·)：a vector space (under the operations of addition and
scalar multiplication defined in V)
Þ W is a subspace of V
 Trivial subspace:
Every vector space V has at least two subspaces.
(1) Zero vector space {0} is a subspace of V.
(2) V is a subspace of V.

24. • Thm 4.5: (Test for a subspace)
If W is a nonempty subset of a vector space V, then W is
a subspace of V if and only if the following conditions hold.
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(1) If u and v are in W, then u+v is in W.
(2) If u is in W and c is any scalar, then cu is in W.

25. • Ex: Subspace of R3
(1) {0} 0 = (0, 0, 0)
(2) Lines through the origin
25/107
 Ex: Subspace of R2
(1) {0} 0 = (0, 0)
(2) Lines through the origin
(3) R2
(3) Planes through the origin
(4) R3

26.  Ex 2: (A subspace of M2×2)
Let W be the set of all 2×2 symmetric matrices. Show that
W is a subspace of the vector space M2×2, with the standard
operations of matrix addition and scalar multiplication.
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Sol:
: vector sapces 2´2 2´2 W Í M M
Let ( ) 1 2 1 1 2 2 A , A ÎW AT = A , AT = A
( ) 1 2 1 2 1 2 1 2 A ÎW, A ÎW Þ A + A T = AT + AT = A + A
k ÎR, AÎW Þ(kA)T = kAT = kA
2 2 is a subspace of ´ W M
( ) 1 2 A + A ÎW
(kAÎW)

27.  Ex 3: (The set of singular matrices is not a subspace of M2×2)
Let W be the set of singular matrices of order 2. Show that
W is not a subspace of M2×2 with the standard operations.
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ù
úû
ù
é
Î = 1 0
é
=
0 0
Sol:
W B , W A Î úû
ù
é
1 0
êë
êë
0 1
0 0
W B A Ï úû
êë
+ =
0 1
2 2 2 is not a subspace of ´ W M

28.  Ex 4: (The set of first-quadrant vectors is not a subspace of R2)
{( , ) : 0 and 0} 1 2 1 2 W = x x x ³ x ³
Show that , with the standard
operations, is not a subspace of R2.
Sol:
(-1)u = (-1)(1, 1) = (-1, -1)ÏW (not closed under scalar
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Let u = (1, 1)ÎW
W is not a subspace of R2
multiplication)

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 Ex 6: (Determining subspaces of R2)
Which of the following two subsets is a subspace of R2?
(a) The set of points on the line given by x+2y=0.
(b) The set of points on the line given by x+2y=1.
Sol:
(a) W = {(x, y) x + 2y = 0} = {(-2t,t) tÎR}
v = (- t t ) ÎW v = (- t t ) ÎW 1 1 1 2 2 2 Let 2 , 2 ,
v + v = (- (t + t ),t + t )ÎW 1 2 1 2 1 2  2
kv = (- (kt ),kt )ÎW 1 1 1 2
W is a subspace of R2
(closed under addition)
(closed under scalar multiplication)
Elementary Linear Algebra: Section 4.3, p.202

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W = {( x, y) x + 2y =1}
Let v = (1,0)ÎW
(-1)v = (-1,0)ÏW
W is not a subspace of R2
(b) (Note: the zero vector is not on the
line)
Elementary Linear Algebra: Section 4.3, p.203

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 Ex 8: (Determining subspaces of R3)
Which of the following subsets is a subspace of ?
{ }
W = x x x x Î
R
(a) ( , ,1) ,
1 2 1 2
W { x x x x x x R}
R
= + Î
1 1 3 3 1 3
3
(b) ( , , ) ,
Sol:
(a) Let v = (0,0,1)ÎW
Þ(-1)v = (0,0,-1)ÏW
W is not a subspace of R3
(b) Let = (v , v + v , v )ÎW, = (u , u + u , u )ÎW 1 1 3 3 1 1 3 3 v u
(v u ,(v u ) (v u ), v u ) W 1 1 1 1 3 3 3 3 v + u = + + + + + Î
( v ,( v ) ( v ), v ) W 1 1 3 3 kv = k k + k k Î
W is a subspace of R3
Elementary Linear Algebra: Section 4.3, p.204

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 Thm 4.6: (The intersection of two subspaces is a subspace)
V W U
If and are both subspaces of a vector space ,
then the intersection of and (denoted by )
U
is also a subspace of .
V W V Ç
U
Elementary Linear Algebra: Section 4.3, p.202