Liner algebra-vector space-1 introduction to vector space and subspace
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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.
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Liner algebra-vector space-1 introduction to vector space and subspace
- 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