This document discusses the key differences between scalar and vector quantities. Scalars only have magnitude, while vectors have both magnitude and direction. It then defines vector spaces as sets of vectors that are closed under vector addition and scalar multiplication. Examples of vector spaces include n-dimensional spaces, matrix spaces, polynomial spaces, and function spaces. Subspaces are also introduced as vector spaces that are subsets of a larger vector space and satisfy the same properties.

2. KEY POINTS TO KNOW
Field
Difference between scalar & vector
Binary Operation

3. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
 A SCALAR QUANTITY IS A QUANTITY
THAT HAS ONLY MAGNITUDE
 IT IS ONE DIMENTIONAL
 ANY CHANGE IN SCALAR QUANTITY IS
THE REFLECTION OF CHANGE IN
MAGNITUDE.
 EXAMPLES:-
MASS,LENGTH,AREA,VOLUME,PRESS
URE,TEMPERATURE,ENERGY,WORK,
PPOWER,TIME,…
 AVECTOR QUANTITY IS A QUANTITY
THAT HAS BOTH MAGNITUDE AND
DIRECTION
 IT CAN BE 1-D,2-D OR 3-D
 ANY CHANGE INVECTOR QUANTITY
ISTHE REFLECTION OF CHANGE IN
EITHER MAGNITUDE OR DIRECTION
OR BOTH.
 EXAMPLES:-
DISPACEMENT,ACCELARATION,
VELOCITY,MOMENTAM,FORCE,…

4. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES



5. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE
(if we add two vectors, we get vector belonging to same space)
(if we multiply a vector by scalar says a real number, we still get a vector)
Vector Space+ .

6. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE

7. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE

8. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟏 𝟐 𝟐, 𝟑 𝟑, 𝟒 𝟒)
𝟏 𝟐 𝟑 𝟒 .
𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟏 𝟐 𝟐, 𝟑 𝟑, 𝟒 𝟒

9. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 𝟒
𝟏 𝟏 𝟐 𝟐, 𝟑 𝟑, 𝟒 𝟒)
𝟏 𝟐 𝟑 𝟒
𝟏 𝟐 𝟑 𝟒

10. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE
Let u=(2, – 1, 5, 0), v=(4, 3, 1, – 1), and w=(– 6, 2, 0, 3) be
vectors in R
4
. Solve for x in 3(x + w) = 2u – v + x
     
 4,,,9
,0,3,9,,,20,5,1,2
322
323
233
2)(3
2
9
2
11
2
9
2
1
2
1
2
3
2
3
2
1









wvux
wvux
wvuxx
xvuwx
xvuwx

11. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
INTERNAL & EXTENAL COMPOSITIONS

12. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE- dEfINITION
-

13. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
N - dIMENSIONAL VECTOR SPACE
An ordered n-tuple: it is a sequence of n-Real numbers
𝟏 𝟐 𝟑 𝒏
𝒏
: the set of all ordered n-tuple
Examples:-
𝟏
𝟏 𝟐 𝟑
2
𝟏 𝟐 𝟏 𝟑 𝟐 𝟐
𝟑
𝟏 𝟐 𝟑 𝟏 𝟐 𝟒 𝟐 𝟑 𝟒
𝟒
𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟒 𝟓 𝟐 𝟑 𝟒 𝟓

14. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
N - dIMENSIONAL VECTOR SPACE
a point
 21, xx
a vector
 21, xx
 0,0
(2) An n-tuple can be viewed as a vector
in Rn with the xi’s as its components.
 21 , xx
 21 , xx
(1) An n-tuple can be viewed as a point in R
n
with the xi’s as its coordinates.
 21 , xx

15. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
Matrix space: (the set of all m×n matrices with real values)nmMV 
Ex: ：(m = n = 2)




















22222121
12121111
2221
1211
2221
1211
vuvu
vuvu
vv
vv
uu
uu












2221
1211
2221
1211
kuku
kuku
uu
uu
k
vector addition
scalar multiplication
METRIX SPACE

16. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
POLYNOMIAL & fUNTIONAL SPACE
n-th degree polynomial space:
(the set of all real polynomials of degree n or less)
)(xPV n
n
nn xbaxbabaxqxp )()()()()( 1100  
n
n xkaxkakaxkp  10)(
)()())(( xgxfxgf 
Function space:
(the set of all real-valued continuous functions defined on the
entire real line.)
),(  cV
)())(( xkfxkf 

17. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
NULL SPACE (OR) zERO VECTOR SPACE

18. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES
1.
2.
3.
4.
5.
6.

19. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES
.
( + ) ------(1) as + =
= + -------(2) as
(1) &(2)

20. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES

21. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE-PROPERTIES

22. MANIKANTA SATYALA || || VECTOR SPACESMANIKANTA SATYALA || || VECTOR SPACES
VECTOR SPACE

23. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE

24. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE

25. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE
fficient

26. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE
fficient

27. VECTOR SUb-SPACE

28. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE

29. MANIKANTA SATYALA || || VECTOR SPACES
VECTOR SUb-SPACE
The set W of ordered triads (x,y,0)
where x , y F is a subspace