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ENGG2013 Unit 14 Subspace and dimension

ENGG2013 Unit 14 Subspace and dimension. Mar, 2011. Yesterday. Every basis in contains two vectors Every basis in contains three vectors. y. x. z. y. x. Basis: Definition. For any given vector in

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ENGG2013 Unit 14 Subspace and dimension

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  1. ENGG2013 Unit 14Subspace and dimension Mar, 2011.

  2. Yesterday • Every basis in contains two vectors • Every basis in contains three vectors y x z y x ENGG2013

  3. Basis: Definition • For any given vector in if there is one and only one choice for the coefficients c1, c2, …,ck, such that we say that these k vectors form a basis of . ENGG2013

  4. Review of set and subset Cities in China Tianjing Beijing Wuhan Shanghai Guangzhou Shenzhen Hong Kong Subset of cities in Guangdongprovince ENGG2013

  5. Review: Intersection and union A union B = {cherry, apple, raspberry, watermelon} F: Set of fruits A intersect B = {raspberry} A: subset of fruit with red skin B: seedless ENGG2013

  6. Subspace: definition • A subspace W in is a subset which is • Closed under addition • Closed under scalar multiplication W ENGG2013

  7. Conceptual illustration W ENGG2013

  8. Example of subspace • The z-axis z y x ENGG2013

  9. Example of subspace • The x-y plane z y x ENGG2013

  10. Non-example • Parabola y x ENGG2013

  11. Intersection • Intersection of two subpaces is also a subspace. z y x For example, the intersectionof the x-y plane and the x-z plane is the same as the x-axis ENGG2013

  12. Union • Union of two subspace is in general not a subspace. • It is closed under scalar multiplicationbut not closed under addition. z y x For example, the unionof the x-y plane and the z axis is not closed under addition ENGG2013

  13. Lattice points • The set is not a subspace • It is closed under addition, • But not closed under scalar multiplication 2 1 1 2 ENGG2013

  14. Subspace, Basis and dimension • Let W be a subspace in • For any given vector in W, if there is one and only one choice for the coefficients c1, c2, …,ck, such that we say that these k vectors form a basis of W. and define the dimension of subspace W by dim(W)=k. ENGG2013

  15. Alternate definition • A set of k vectors is called a basis of a subspace W in , if • The k vectors are linearly independent • The span of them is W. The dimension of W is defined as k. We say that W is generated by these k vectors. ENGG2013

  16. Example • Let W be the x-z plane • W is a subspace • u and v form a basisof W. • The dimension of W is 2. z y W x ENGG2013

  17. Example • Let W be the y-axis • The set containing only one elementis a basis of W. Dimension of W is 1. z y W x ENGG2013

  18. Question • Let W be the y-axisshifted to the right by one unit. • What is the dimensionof W? z y W 1 x ENGG2013

  19. Question • Let W be the straight line x=y=z. • What is the dimension of W? ENGG2013

  20. Question • Find a basis for the plane ENGG2013

  21. Question • Find a basis for the intersection of (This is the intersection of two planes: x – 2y – z = 0, and x + y + z = 0.) ENGG2013

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