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

2. Yesterday Every basis in contains two vectors Every basis in contains three vectors kshumENGG20132 x y x y z

3. Basis: Definition For any given vector in if there is one and only one choice for the coefficients c 1, c 2, …,c k, such that we say that these k vectors form a basis of. kshumENGG20133

4. Review of set and subset kshumENGG20134 Cities in China Shanghai Beijing Hong Kong Tianjing Wuhan Guangzhou Shenzhen Subset of cities in Guangdong province

5. Review: Intersection and union kshumENGG20135 F: Set of fruits A: subset of fruit with red skin B: seedless A union B = {cherry, apple, raspberry, watermelon} A intersect B = {raspberry}

6. Subspace: definition A subspace W in is a subset which is – Closed under addition – Closed under scalar multiplication kshumENGG20136 W

7. Conceptual illustration kshumENGG20137 W

8. Example of subspace The z-axis kshumENGG20138 x y z

9. Example of subspace The x-y plane kshumENGG20139 x y z

10. Non-example Parabola kshumENGG201310 x y

11. Intersection Intersection of two subpaces is also a subspace. kshumENGG201311 x y z For example, the intersection of the x-y plane and the x-z plane is the same as the x-axis

12. Union Union of two subspace is in general not a subspace. – It is closed under scalar multiplication but not closed under addition. kshumENGG201312 x y z For example, the union of the x-y plane and the z axis is not closed under addition

13. Lattice points The set is not a subspace – It is closed under addition, – But not closed under scalar multiplication kshumENGG201313 1 1 2 2

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 c 1, c 2, …,c k, such that we say that these k vectors form a basis of W. and define the dimension of subspace W by dim(W)=k. kshumENGG201314

15. Alternate definition A set of k vectors is called a basis of a subspace W in, if 1.The k vectors are linearly independent 2.The span of them is W. The dimension of W is defined as k. We say that W is generated by these k vectors. kshumENGG201315

16. Example Let W be the x-z plane W is a subspace u and v form a basis of W. The dimension of W is 2. kshumENGG201316 x y z W

17. Example Let W be the y-axis The set containing only one element is a basis of W. Dimension of W is 1. kshumENGG201317 x y z W

18. Question Let W be the y-axis shifted to the right by one unit. What is the dimension of W? kshumENGG201318 x y z W 1

19. Question Let W be the straight line x=y=z. What is the dimension of W? kshumENGG201319

20. Question Find a basis for the plane kshumENGG201320

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.) kshumENGG201321