ENGG2013 Unit 15 Rank, determinant, dimension, and the links between them

1. ENGG2013 Unit 15Rank, determinant, dimension,and the links between them Mar, 2011.

2. Review on “rank” • “row-rank of a matrix” counts the max. number of linearly independent rows. • “column-rank of a matrix” counts the max. number of linearly independent columns. • One application: Given a large system of linear equations, count the number of essentially different equations. • The number of essentially different equations is just the row-rank of the augmented matrix. ENGG2013

3. Evaluating the row-rank by definition Linearly independent Linearly independent Linearly independent Linearly dependent Linearly independent Linearly independent Linearly dependent  Row-Rank = 2 ENGG2013

4. Calculation of row-rank via RREF Row reductions Row-rank = 2 Row-rank = 2 Because row reductionsdo not affect the numberof linearly independent rows ENGG2013

5. Calculation of column-rank by definition List all combinationsof columns  Column-Rank = 2 Linearly independent?? Y Y Y Y Y Y Y Y Y Y N N N N N ENGG2013

6. Theorem Given any matrix, its row-rank and column-rank are equal. In view of this property, we can just say the “rank of a matrix”. It means either the row-rank of column-rank. ENGG2013

7. Why row-rank = column-rank? • If some column vectors are linearly dependent, they remain linearly dependent after any elementary row operation • For example, are linearly dependent ENGG2013

8. Why row-rank = column-rank? • Any row operation does not change the column- rank. • By the same argument, apply to the transpose of the matrix, we conclude that any column operation does not change the row-rank as well. ENGG2013

9. Why row-rank = column-rank? Apply row reductions.row-rank and column-rankdo not change. Apply column reductions.row-rank and column-rankdo not change. The top-left corner isan identity matrix. The row-rank and column-rank of this “normal form” is certainlythe size of this identity submatrix, and are therefore equal. ENGG2013

10. DISCRIMINANT, DETERMINANT AND RANK ENGG2013

11. Discriminant of a quadratic equation • y = ax2+bx+c • Discirminant of ax2+bx+c = b2-4ac. • It determines whether the roots are distinct or not y x ENGG2013

12. Discriminant measures the separation of roots • y = x2+bx+c. Let the roots be  and . • y = (x – )(x – ). Discriminant = ( – )2. • Discriminant is zero means that the two roots coincide. y ( – )2 x ENGG2013

13. Discriminant is invariant under translation • If we substitute u= x – t into y = ax2+bx+c, (t is any real constant), then the discriminant of a(u+t)2+b(u+t)+c, as a polynomial in u, is the same as before. y ( – )2 u ENGG2013

14. Determinant of a square matrix • The determinant of a square matrix determine whether the matrix is invertible or not. • Zero determinant: not invertible • Non-zero determinant: invertible. ENGG2013

15. Determinant measure the area • 22 determinant measures the area of a parallelogram. • 33 determinant measures the volume of a parallelopiped. • nn determinant measures the “volume” of some “parallelogram” in n-dimension. • Determinant is zero means that the columns vectors lie in some lower-dimensional space. ENGG2013

16. Determinant is invariant under shearing action • Shearing action = third kind of elementary row or column operation ENGG2013

17. Rank of a rectangular matrix • The rank of a matrix counts the maximal number of linearly independent rows. • It also counts the maximal number of linearly independent columns. • It is an integer. • If the matrix is mn, then the rank is an integer between 0 and min(m,n). ENGG2013

18. Rank is invariant under row and column operations Rank = 2 Rank = 2 Rank = 2 Rank = 2 Rank = 2 Rank = 2 Rank = 2 ENGG2013

19. Comparison between det and rank Determinant Rank Integer Defined to any rectangular matrix When applied to nn square matrix, rank=n implies existence of inverse. • Real number • Defined to square matrix only • Non-zero det implies existence of inverse. • When det is zero, we only know that all the columns (or rows) together are linearly dependent, but don’t know any information about subset of columns (or rows) which are linearly independent. ENGG2013

20. 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

21. Yet another interpretation of rank • Recall that a subspace W in is a subset which is • Closed under addition: Sum of any two vectors in W stay in W. • Closed under scalar multiplication: scalar multiple of any vector in W stays in W as well. W ENGG2013

22. Closedness property of subspace W ENGG2013

23. Geometric picture x – 3y + z = 0 W z y x W is the planegenerated, or spanned,by these vectors. ENGG2013

24. Basis and dimension • A basis of a subspace W is a set of linearly independent vectors which span W. • A rigorous definition of the dimension is: Dim(W) = the number of vectors in a basis of W. z y W x ENGG2013

25. Rank as dimension • In this context, the rank of a matrix is the dimension of the subspace spanned by the rows of this matrix. • The least number of row vectors required to span the subspace spanned by the rows. • The rank is also the dimension of the subspace spanned by the column of this matrix. • The least number of column vectors required to span the subspace spanned by the columns ENGG2013

26. Example x – 2y + z = 0 z y Rank = 2 The three row vectorslie on the same plane. Two of them is enoughto describe the plane. x ENGG2013

27. INTERPOLATION ENGG2013

28. Polynomial interpolation • Given n points, find a polynomial of degree n-1 which goes through these n points. • Technical requirements: • All x-coordinates must be distinct • y-coordinates need not be distinct. y3 y2 y4 y1 x1 x2 x3 x4 ENGG2013

29. Lagrange interpolation • Lagrange interpolating polynomial for four data points: ENGG2013

30. Computing the coefficients by linear equations • We want to solve for coefficients c3, c2, c1, and c0, such that or equivalently ENGG2013

31. The theoretical basis for polynomial interpolation • The determinant of a vandermonde matrix is non-zero, if all xi’s are distinct. Hence, we can always find the matrix inverse and solve the system of linear equations. ENGG2013