ENGG2013 Unit 15

Rank, determinant, dimension,and the links between them

Mar, 2011.

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 2

Evaluating the row-rank by definition

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Linearly independent

Linearly independent

Linearly independent

Linearly independent

Linearly independent

Linearly dependent

Linearly dependent Row-Rank = 2

Calculation of row-rank via RREF

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Row reductions

Row-rank = 2Row-rank = 2

Because row reductionsdo not affect the numberof linearly independent rows

Calculation of column-rank by definition

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List all combinationsof columns

Linearly independent??

Y Y Y Y Y Y Y Y Y

N N

Y

N N

N

Column-Rank = 2

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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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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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.

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Why row-rank = column-rank?

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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.

DISCRIMINANT, DETERMINANT AND RANK

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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

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x

y

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.

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x

y

( – )2

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.

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u

y

( – )2

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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Determinant measure the area

• 22 determinant measures the area of a parallelogram.

• 33 determinant measures the volume of a parallelopiped.

• nn determinant measures the “volume” of some “parallelogram” in n-dimension.

• Determinant is zero means that the columns vectors lie in some lower-dimensional space.

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Determinant is invariant under shearing action

• Shearing action = third kind of elementary row or column operation

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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 mn, then the rank is an

integer between 0 and min(m,n).

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Rank is invariant under row and column operations

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Rank = 2

Rank = 2

Rank = 2

Rank = 2

Rank = 2Rank = 2Rank = 2

Comparison between det and rank

Determinant• 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.

Rank• Integer• Defined to any rectangular

matrix• When applied to nn

square matrix, rank=n implies existence of inverse.

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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 . kshum ENGG2013 20

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.

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W

Closedness property of subspace

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W

Geometric picture

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x

y

zW

W is the planegenerated, or spanned,by these vectors.

x – 3y + z = 0

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.

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x

yz

W

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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Example

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x

y

z

The three row vectorslie on the same plane.

Two of them is enoughto describe the plane.

x – 2y + z = 0

Rank = 2

INTERPOLATION

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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.

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x1 x2 x3 x4

y1

y2y3

y4

Lagrange interpolation

• Lagrange interpolating polynomial for four data points:

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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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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.

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