1. ENGG2013 Unit 13 Basis Feb, 2011.

2. Question 1 Find the value of c 1 and c 2 such that kshumENGG20132

3. Question 2 Find the value of c 1 and c 2 such that kshumENGG20133

4. Question 3 Find c 1, c 2, c 3 and c 4 such that kshumENGG20134

5. 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. kshumENGG20135

6. Example form a basis of. Another notation is: is a basis of. kshumENGG20136 1 1

7. Example form a basis of. Another notation is: is a basis of. kshumENGG20137 2 2

8. Non-Example is not a basis of. kshumENGG20138 1 1

9. Alternate definition of basis A set of k vectors is a basis of if the k vectors satisfy: 1.They are linear independent 2.The span of them is equal to (this is a short-hand of the statement that: every vector in can be written as a linear combination of these k vectors.) kshumENGG20139

10. More examples is a basis of kshumENGG201310 3 3

11. Question Is a basis of kshumENGG201311 1 1 x y z

12. Question Is a basis of ? kshumENGG201312 1 1 x y z 1

13. Question Is a basis of ? kshumENGG201313 3 2 x y z 1 1

14. Question Is a basis of ? kshumENGG201314 2 x y z 1 1

15. Question Is a basis of ? kshumENGG201315 2 x y z 1 1

16. Fact Any two vectors in do not form a basis. – Because they cannot span the whole. Any four or more vectors in do not form a basis – Because they are not linearly independent. We need exactly three vectors to form a basis of. kshumENGG201316

17. A test based on determinant Somebody gives you three vectors in. Can you tell quickly whether they form a basis? kshumENGG201317

18. Theorem Three vectors in form a basis if and only if the determinant obtained by writing the three vectors together is non-zero. Proof:  Let the three vectors be Assume that they form a basis. In particular, they are linearly independent. By definition, this means that if then c 1, c 2, and c 3 must be all zero. By the theorem in unit 12 (p.17), the determinant is nonzero. kshumENGG201318 This theorem generalizes to higher dimension naturally. Just replace 3x3 det by nxn det

19. The direction  of the proof In the reverse direction, suppose that We want to show that 1.The three columns are linearly independent 2.Every vector in can be written as a linear combination of these three columns. kshumENGG201319

20. The direction  of the proof 1.Linear independence: Immediate from the theorem in unit 12 (8  3). 2.Let be any vector in. We want to find coefficients c 1, c 2 and c 3 such that Using (8  1), we know that we can find a left inverse of. We can multiply by the left inverse from the left and calculate c 1, c 2, c 3. kshumENGG201320

21. Example Determine whether form a basis. Check the determinant of kshumENGG201321

22. Summary A basis of contains the smallest number of vectors such that every vector can be written as a linear combination of the vectors in the basis. Alternately, we can simply say that: A basis of is a set of vectors, with fewest number of vectors, such that the span of them is. kshumENGG201322