The document discusses vector spaces and subspaces. It defines vectors in Rn as n-tuples of real numbers and describes operations on vectors like addition and scalar multiplication. A vector space is a set with vectors that is closed under these operations and satisfies other axioms. Examples given include Rn, the space of matrices, and polynomial spaces. A subspace is a subset of a vector space that itself is a vector space under the operations in the larger space.

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)
2
/10
7

3. 4.1 VECTORS IN RN
3
/10
7
 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
/10
7
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.
5
/10
7
 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)
6
/10
7
 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.

7. 7
/10
7
 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.
8
/10
7
(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
9
/10
7
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).

10. 10
/10
7
(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
11
/10
7

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:
13
/10
7
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
[ , , , ]
14
/10
7
( , , , )
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

16. 16
/10
7
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.
24
/10
7
(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.
26
/10
7
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.
27
/10
7
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
28
/10
7
Let u = (1, 1)ÎW
W is not a subspace of R2
multiplication)

29. 29
/10
7
 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

30. 30
/10
7
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

31. 31
/10
7
 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

32. 32
/10
7
 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