1. ENGG2013 Unit 15 Rank, 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.
kshum
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
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ENGG2013

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

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

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

10. DISCRIMINANT, DETERMINANT AND RANK
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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
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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
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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
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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.
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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.
kshum
ENGG2013

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

18. Rank is invariant under row and column operations
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ENGG2013

19. Comparison between det and rank
Determinant
Rank
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.
Integer
Defined to any rectangular matrix
When applied to nn square matrix, rank=n implies existence of inverse.
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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
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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
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ENGG2013

22. Closedness property of subspaceW
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ENGG2013

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

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

27. INTERPOLATION
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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
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ENGG2013

29. Lagrange interpolation
Lagrange interpolating polynomial for four data points:
\begin{align*}
f(x)&=y_1\frac{(x-x_2)(x-x_3)(x-x_4)}{(x_1-x_2)(x_1-x_3)(x_1-x_4)}+
y_2\frac{(x-x_1)(x-x_3)(x-x_4)}{(x_2-x_1)(x_2-x_3)(x_2-x_4)}+ \\
& y_3\frac{(x-x_1)(x-x_2)(x-x_4)}{(x_3-x_1)(x_3-x_2)(x_3-x_4)}+
y_4\frac{(x-x_1)(x-x_2)(x-x_3)}{(x_4-x_1)(x_4-x_2)(x_4-x_3)}
\end{align*}
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ENGG2013

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