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ENGG2013 Unit 13 Basis

ENGG2013 Unit 13 Basis. Feb, 2011. Question 1. Find the value of c 1 and c 2 such that. Question 2. Find the value of c 1 and c 2 such that. Question 3. Find c 1 , c 2 , c 3 and c 4 such that. Basis: Definition. For any given vector in

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ENGG2013 Unit 13 Basis

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  1. ENGG2013 Unit 13Basis Feb, 2011.

  2. Question 1 • Find the value of c1 and c2 such that ENGG2013

  3. Question 2 • Find the value of c1 and c2 such that ENGG2013

  4. Question 3 • Find c1, c2,c3 and c4 such that ENGG2013

  5. 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

  6. Example • form a basis of . • Another notation is: is a basis of . 1 1 ENGG2013

  7. Example • form a basis of . • Another notation is: is a basis of . 2 2 ENGG2013

  8. Non-Example • is not a basis of . 1 1 ENGG2013

  9. Alternate definition of basis • A set of k vectors is a basis of if the k vectors satisfy: • They are linear independent • 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.) ENGG2013

  10. More examples • is a basis of 3 3 ENGG2013

  11. Question • Is a basis of z 1 1 x y ENGG2013

  12. Question • Is a basis of ? z 1 1 1 x y ENGG2013

  13. Question • Is a basis of ? z 1 1 3 x 2 y ENGG2013

  14. Question • Is a basis of ? z 2 1 x 1 y ENGG2013

  15. Question • Is a basis of ? z 2 1 x 1 y ENGG2013

  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 . ENGG2013

  17. A test based on determinant • Somebody gives you three vectors in . • Can you tell quickly whether they form a basis? ENGG2013

  18. This theorem generalizesto higher dimension naturally.Just replace 3x3 det by nxn det 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 c1, c2, and c3 must be all zero. By the theorem in unit 12 (p.17) , the determinant is nonzero. ENGG2013

  19. The direction  of the proof • In the reverse direction, suppose that • We want to show that • The three columns are linearly independent • Every vector in can be written as a linear combination of these three columns. ENGG2013

  20. The direction  of the proof • Linear independence: Immediate from the theorem in unit 12 (8  3). • Let be any vector in . We want to find coefficients c1, c2 and c3 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 c1, c2, c3. ENGG2013

  21. Example • Determine whether form a basis. • Check the determinant of ENGG2013

  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 . ENGG2013

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