CSIR-UCG National Eligibility Test (NET) for Junior ...mathskthm.6te.net/Linear Algebra 1.pdf · CSIR-UCG National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship

CSIR-UCG National Eligibility Test (NET) for Junior Research Fellowship and Lecturer-ship COMMON SYLLABUS FOR PART ‘B’ AND ‘C’ MATHEMATICAL SCIENCES UNIT-1 Analysis: Elementary set theory, finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property, supremum, infimum.

Sequences and series, convergence, limsup, liminf.

Bolzano Weierstrass theorem, Heine Borel theorem.

Continuity, uniform continuity, differentiability, mean value theorem.

Sequences and series of functions, uniform convergence.

Riemann sums and Riemann integral, Improper Integrals.

Monotonic functions, types of discontinuity, fuctions of bounded variation, Lebesgue measure, Lebesgue integral.

Functions of several variables, directional derivatives, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.

Metric spaces, compactness, connectedness, Normed linear spaces. Spaces of continuous functions as examples.

Linear Algebra: Vector spaces, subspaces, linearly dependence, basis, dimension, algebra of linear transformations.

Algebra of matrices, rank and determinant of matrices, linear equations.

Eigenvalues and eigenvectors, Cayley-Hamilton theorem.

Matrix representation of linear transformations. Change of basis, canonical forms, diagonal forms, triangular forms and Jordan forms.

Inner product spaces, ortho-normal basis.

Quadratic forms, reduction and classification of quadratic forms.

1) Matrices Definition: MATRIX

A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix.

Some examples of examples Matrix

A general 푚 × 푛 is denoted by 퐴 = 푎

Definition: Diagonal Matrix

A matrix A with n rows and n columns is called a square matrix of order n, and the shaded entries 푎 , 푎 푎 , …, 푎 in the matrix are said to be on the main diagonal of A.

2) Operations On Matrices 1) Equality 2) Sum or Addition 3) Subtraction or Difference 4) Product or Matrix Multiplication

A) Scalar Multiplication 푐퐴 = 푐푎

B) Matrix Multiplication 퐴 = 푎 퐵 = 푏 The (i ,j)th entry (퐴퐵) of 퐴퐵 is

Consider the following example

Note that before taking the product first determining whether a product Is defined

3) Partitioned Matrices A matrix can be subdivided or partitioned into smaller matrices by inserting

horizontal and vertical rules between selected rows and columns. For example, the following are three possible partitions of a general 3 × 4 matrix A—the first is a partition of A into four sub matrices 퐴 , 퐴 퐴 , 퐴 and ; the second is a partition of A into its row matrices 퐫 , 퐫 , 퐫 ; and the third is a partition of A into its column matrices 퐜 , 퐜 , 퐜 and 퐜 :

4) Matrix Multiplication By Columns And By Rows

Sometimes it may be desirable to find a particular row or column of a matrix product 퐴퐵without computing the entire product. The following results, whose proofs are left as exercises, are useful for that purpose:

Consider the Example with Matrices 퐴 and B

Multiplication by columns

Multiplication by rows

5) Linear Combination Of Matrices

If 퐴 ,퐴 , … ,퐴 are matrices of the same size and 푐 , 푐 , … , 푐 are scalars, then an expression of the form

푐 퐴 + 푐 퐴 + … 푐 퐴

is called a linear combination of 퐴 ,퐴 , … ,퐴 with coefficients 푐 , 푐 , … , 푐 . For example, if A, B, and C are the matrices

6) Matrix Products As Linear Combinations

Row and column matrices provide an alternative way of thinking about matrix multiplication. For example, suppose that

Then

The product of a matrix 퐴푥 with a column matrix 풙 is a linear combination of

the column matrices of A with the coefficients coming from the matrix 풙.

We can show that the product 푦퐴 of a 1 × 푚 matrix y with an 푚 × 푛 matrix A

is a linear combination of the row matrices of A with scalar coefficients coming from y.

Linear Combinations The matrix product

can be written as the linear combination of column matrices

The matrix product

can be written as the linear combination of row matrices

Columns of a Product 푨푩 as Linear Combinations

The column matrices of 푨푩 can be expressed as linear combinations of the column matrices of A as follows:

7) Matrix Form Of A Linear System Matrix multiplication has an important application to systems of linear equations. Consider any system of m linear equations in n unknowns.

Since two matrices are equal if and only if their corresponding entries are equal, we can replace the m equations in this system by the single matrix equation

The 푚 × 1 matrix on the left side of this equation can be written as a product to give If we designate these matrices by A, x, and b, respectively, then the original system of m equations in n unknowns has been replaced by the single matrix equation

푨풙 = 풃 The 푚 × 푛 matrix A in this equation is called the coefficient matrix of the system, the 1 × 푛 matrix 풙 is called variable matrix, the 1 × 푚 matrix 풙 is called constant matrix The augmented matrix for the system is obtained by adjoining b to A as the last column; thus the augmented matrix is

8) Matrices Defining Functions The equation푨풙 = 풃 with 푨 and 풃 given defines a linear system to be solved

for x. But we could also write this equation as

풚 = 푨풙

Where 푨 and 풙 are given. In this case, we want to compute 풚 . If 푨 is 푚 × 푛,

then this is a function that associates with every 푛 × ퟏ column vector 풙 an 푚 × ퟏ

column vector 풚 and we may view 푨 as defining a rule that shows how a given 풙

is mapped into a corresponding 풚.

A Function Using Matrices

Consider the following matrices.

The product 풚 = 푨풙 is

so the effect of multiplying A by a column vector is to change the sign of the second entry of the column vector. For the matrix The product 풚 = 푩풙 is

so the effect of multiplying B by a column vector is to interchange the first and second entries of the column vector, also changing the sign of the first entry.

Geometric interpretation

If we view the column vector x as locating a point (푎, 푏) in the plane, then the effect of A is to reflect the point about the x-axis (Figure a)

Whereas the effect of B is to rotate the line segment from the origin to the point

through a right angle (Figure b).

Scaling, Uniform scaling Shearing Rotation Reflection

9) Transpose Of A Matrix If A is any 푚 × 푛, matrix, then the transpose of A, denoted by 퐴 or 퐴 is

defined to be the 푛 × 푚, matrix that results from interchanging the rows and columns of A; that is, the first column of by 퐴 is the first row of A, the second column of by 퐴 is the second row of A, and so on.

Some Transposes

10) Trace Of A Matrix If 퐴 is a square matrix, then the trace of 퐴, denoted by 푡푟퐴, is defined to

be the sum of the entries on the main diagonal of 퐴. The trace of A is

undefined if A is not a square matrix.

Properties of trace Let 퐴,퐵 be square matrices of order 푛, then

(a) 푡푟(퐴 + 퐵) = 푡푟(퐴) + 푡푟(퐵)

(b) 푡푟(퐴퐵) = 푡푟(퐵퐴)

(c) 푡푟(푘퐴) = 푘. 푡푟(퐴); 푘 푖푠 푠푐푎푙푎푟

(d) 푡푟(퐴 ) = 푡푟(퐴)

Example10.1: A matrix B is said to be a square root of a matrix A if 퐵퐵 = 퐴 . (a) Find two square roots of 퐴 = 2 2

2 2

(b) How many different square roots can you find of 퐴 = 5 00 9 ?

(c) Do you think that every matrix has at least one square root? Explain your reasoning.

Solution :(a) Let 퐵 = 푎 푏

푐 푑 . Then 퐵 = 퐴 implies that

푎 + 푏푐 = 2 … [1]푎푏 + 푏푑 = 2 … [2]푎푐 + 푐푑 = 2 … [3]푏푐 + 푑 = 2 … [4]

⎭⎬

⎫ Set of equations obtained from 퐵 = 퐴

One might note that 푎 = 푏 = 푐 = 푑 = 1 and 푎 = 푏 = 푐 = 푑 = – 1

satisfy eqns.

Solving the eqn [1] and [4] of the above equations simultaneously yields

푎 = 푑 .

Thus a = ±d.

Solving the remaining 2 equations yields

푐(푎 + 푑) = 푏(푎 + 푑) = 2.

Therefore 푎 ≠ –푑 and 푎 and d cannot both be zero.

Hence we have 푎 = 푑 ≠ 0, so that 푎푐 = 푎푏 = 1, or b = c = 1/a.

The equation [1] in the set of eqns then becomes

푎 + 1/푎 = 2 or

푎 – 2푎 + 1 = 0.

Thus 푎 = ±1.

Example10.1: A matrix B is said to be a square root of a matrix A if 퐵퐵 = 퐴 . (a) Find two square roots of 퐴 = 2 2

2 2

(b) How many different square roots can you find of 퐴 = 5 00 9 ?

(c) Do you think that every matrix has at least one square root? Explain your reasoning.

(b) Using the reasoning and the notation of Part (a),

Show that either 푎 = –푑 or 푏 = 푐 = 0.

If 푎 = –푑, then 푎 + 푏푐 = 5 and 푏푐 + 푎 = 9.

This is impossible.

So, we have b = c = 0.

This implies that 푎 = 5, 푑 = 9.

Thus √5 00 3

, −√5 00 3

, √5 00 −3

, −√5 00 −3

.

There are the four square roots of A

Note that if A were 5 00 5 , say, then B =

1 푟−1 would be a square root of A

for every nonzero real number r and there would be infinitely many other

square roots as well.

(c) By an argument similar to the above, show that if, for instance,

퐴 = −1 00 1 and 퐵 = 푎 푏

푐 푑 where BB = A,

Then either a = –d or b = c = 0.

Each of these alternatives leads to a contradiction.

11) Properties Of Matrix Operations A) AB and BA Need Not Be Equal B) C) D) E) F) G) H) I) J) K) L) M)

The Cancellation Law Does Not Hold

Consider the matrices You should verify that As 퐴퐵 = 퐴퐶 ⇏ 퐵 = 퐶 and 퐴 ≠ 0,퐷 ≠ 0 ⟹ 퐴퐷 = 0

12) Invertible Matrix

If A is a square matrix and if a matrix B of the same size can be found such

that 퐴퐵 = 퐵퐴 = 퐼 , then A is said to be invertible (non-singular) and B is

called an inverse of A. If no such matrix B can be found, then A is said to be

singular.

and We write it as 퐵 = 퐴 And 퐴퐴 = 퐴 퐴 = 퐼

Results: 1) If R is the reduced row-echelon form of an n × n, matrix A, then

either R has a row of zeros or R is the identity matrix I ,

2) If B and C are both inverses of the matrix A, then B = C .

3) If A and B are invertible matrices of the same size, then AB is

invertible and (AB) = B A

13) Laws of Exponents

If A is a square matrix and r and s are integers, thenA A = A

If A is an invertible matrix, then:

(a) A is invertible and . (퐴 ) = 퐴

(b) A is invertible and (퐴 ) = (퐴 ) for 푛 = 0, 1, 2 , … .

(c) For any nonzero scalar k, then matrix 푘퐴 is invertible and (푘퐴) = 퐴 .

Powers of a Matrix

Let A and A be

퐴 = (퐴 ) = (퐴 ) 1 21 3

1 21 3 = 11 30

15 41

(퐴) = (퐴 ) = 3 −2−1 1

3 −2−1 1

3 −2−1 1 = 41 −30

−15 11

14) Polynomial Expressions Involving Matrices If A is a square matrix of order 푚 × 푚 and

If is any polynomial then we define

Where I is identity matrix of order 푚 × 푚

Matrix Polynomial

If

Then

Example 13.1 a) Give an example in which

b) Fill in the blank to create a matrix identity that is valid for all choices of A

and B. (퐴 + 퐵) = 퐴 + −− −

Showing That a Matrix Is Not Invertible Consider the matrix

Applying the procedure of row operations

Since we have obtained a row of zeros on the left side, A is not invertible

15) Determinant Definition: A function from set of all square matrices to set of all real numbers or complex numbers such that every square matrix A assigns a unique scalar then the scalar is called determinant of A. it is denoted as |A| or det A.

The 2x2 matrix 퐴 = 푎 푏푐 푑 is invertible if 푎푑 − 푏푐 ≠ 0 then the

expression 푎푑 − 푏푐 it is called determinant of A The inverse of A is obtained by the formula 퐴 =

| |푑 −푏−푐 푎

Minor and Co-factor : Definition:

If 퐴 is a square matrix, then the minor of entry 풂풊풋 is denoted by 푴풊풋 and is

defined to be the determinant of the sub-matrix that remains after the 푖th

row and 푗th column are deleted from A . The number (−ퟏ)풊 풋푴풊풋 is denoted

by 푪풊풋 and is called the cofactor of entry 풂풊풋 .

Let `

Then minor of 푎 = 3 is

The co-factor of 푎 = 3 is

Cofactor Expansions

The definition of a 3 × 3 determinant in terms of minors and cofactors is

This method of evaluating det퐴 is called cofactor expansion along the first row of A. If A is 3 × 3 matrix, then its determinant its determinant in terms of minors and cofactors is

Cofactor expansion of first row , first column , second row, second row, second column , third row, third column respectivly

The determinant of an n × n matrix A can be computed by multiplying the entries in any row (or column) by their cofactors and adding the resulting products; that is, for each 1 ≤ 푖 ≤ 푛 and 1 ≤ 푗 ≤ 푛.

Cofactor expansion along the jth column

det(퐴) = 푎 퐶 + 푎 퐶 + ⋯+ 푎 퐶 Cofactor expansion along the ith row

det(퐴) = 푎 퐶 + 푎 퐶 +⋯+ 푎 퐶

Smart Choice of Row or Column

Let A be 4x4 matrix

Choose 2nd column to evaluate the det A

Determinant by sign elementary Products

푑푒푡푎 푎푎 푎 = 푎11푎22 −푎12푎21

1 2 3−47

5−8

69

Properties of the Determinant function

Let 퐴 be an 푛 × 푛 matrix. The following four properties f the determinant function:

(a) If 퐵 is obtained from 퐴 by interchanging any two rows of 퐴, then |퐵| = −|퐴|. (b) If 퐵 is obtained from 퐴 by multiplying the 푖 row of 퐴 by a non-zero scalar 푘,

then |퐵| = 푘|퐴|. (c) If 퐵 is obtained from 퐴 by adding to the elements of any row of 퐴 푘 multiple

of corresponding elements of another row of 퐴, then |퐵| = |퐴| (d) If 퐴,퐵 are square matrices of the same order, then |퐴퐵| = |퐴|. |퐵|

Using Row Reduction to Evaluate a Determinant

Geometric Interpretation of Determinants

The absolute value of the determinant

a) The determinant of 2X2 Matrix

푑푒푡푢 푢푣 푣 is equal to the area of the parallelogram in 2-space

determined by the vectors푢 = (푢 ,푢 ) and v= (푣 , 푣 ) (See Figure a.)

b) The determinant of 2X2 Matrix

푑푒푡푢 푢 푢푣푤

푣푤

푣푤

is equal to the volume of the parallelepiped in 3-

space determined by the vectors푢 = (푢 ,푢 , 푢 ), 푣 = (푣 , 푣 ,푣 ) and

푤 = (푤 ,푤 ,푤 )

Theorem: Let 퐴,퐵 be 푛 × 푛 matrices which differ in a single row say 푟 row. Let 퐶 be an 푛 × 푛 matrix whose 푟 row is obtained by adding 푟 rows of 퐴 and 퐵 and all other rows of 퐶 are respectively the same as those of 퐴, then

det(퐶) = det(퐴) + det (퐵)

Example: Let 퐴 =1 0 12 −2 35 6 7

and 퐵 =1 0 11 2 35 6 7

. Let 퐶 =1 0 13 0 65 6 7

Then det(퐶) = −18 푎푛푑 det(퐴) = −10, det(퐵) = −8

Remark: For any square matrices 퐴,퐵 of order 푛 × 푛 it is not true that

det(퐴 + 퐵) = det(퐴) + det (퐵)

Example: Let 퐴 = 5 9−1 0 , 퐵 = 1 2

2 3 . Then 퐴 + 퐵 = 6 111 3

Here det(퐴 + 퐵) = 7 푎푛푑 det(퐴) = 9, det(퐵) = −1

Thus det(퐴 + 퐵) ≠ det(퐴) + det(퐵)

Theorem: Let 퐴 be 푛 × 푛 matrix and if 푘is a non zero scalar then

det(푘퐴) + 푘 det(퐴).

16) Diagonal Matrices A square matrix in which all the entries off the main diagonal are zero is called a diagonal matrix. Here are some examples:

A general 푛 × 푛 diagonal matrix D can be written as

A diagonal matrix is invertible if and only if all of its diagonal entries are nonzero; in this case the inverse of D is

One should verify that 퐷퐷 = 퐷 퐷 = 퐼.

Powers of diagonal matrices are easy to compute; we leave it for the reader to verify that if D is the diagonal matrix and k is a positive integer, then

Inverses and Powers of Diagonal Matrices

If

Then

Upper and Lower Triangular Matrices

Properties: The following theorem lists some of the basic properties of triangular matrices.

(a)The transpose of a lower triangular matrix is upper triangular, and the transpose

of an upper triangular matrix is lower triangular.

(b) The product of lower triangular matrices is lower triangular, and the product of

upper triangular matrices is upper triangular.

(c) A triangular matrix is invertible if and only if its diagonal entries are all

nonzero.

(d) The inverse of an invertible lower triangular matrix is lower triangular, and the

inverse of an invertible upper triangular matrix is upper triangular.

17) Symmetric Matrices A square matrix A is called symmetric if 퐴 = 퐴 or

A square matrix 퐴 = 푎×

is called symmetric if 푎 = 푎

If A and B are symmetric matrices with the same size, and if k is any scalar, then:

(a) A is symmetric.

(b) A + B and A − B are symmetric.

(c) kA is symmetric.

Remark It is not true, in general, that the product of symmetric matrices is symmetric.

Let A and B be symmetric matrices with the same size.

(AB) = B A = BA

Usually 퐴퐵 ≠ 퐵퐴

The product of two symmetric matrices is symmetric if and only if the matrices

commute.

Properties of Symmetric Matrices

(A) If A is an invertible symmetric matrix, then 퐴 is symmetric.

Products 퐴퐴 and퐴 퐴:

Matrix products of the form 퐴퐴 and 퐴 퐴 and arise in a variety of applications. If A

is an 푚 × 푛 matrix, then 퐴 is an 푛 × 푚 matrix, so the products 퐴퐴 and퐴 퐴 are both

square matrices; the matrix 퐴퐴 has size 푚 × 푚 , and the matrix 퐴 퐴 has size 푛 × 푛.

Such products are always symmetric since

The Product of a Matrix and Its Transpose Is Symmetric Let A be 2X3 matrix

Then

Properties

(B) If A is an invertible matrix, then 퐴퐴 and 퐴 퐴 are also invertible.

(C) Let A be an 푛 × 푛 symmetric matrix, then (a) 퐴 is symmetric. (b) 2퐴 − 3퐴 + 퐼 is symmetric. (c) 퐴 is symmetric, if k is any nonnegative integer. (d) If 푝(푥) is a polynomial, is 푝(퐴) necessarily symmetric? Explain.

18) Skew-symmetric Matrices A square matrix A is called skew symmetric if 퐴 = − 퐴 or

A square matrix 퐴 = 푎×

is called skew symmetric if 푎 = −푎

Properties

(a) If A is an invertible skew-symmetric matrix, then 퐴 is skew-symmetric. (b) If A and B are skew-symmetric, then so are 퐴 , A + B, A − B and 푘A for any scalar k. (c) Every square matrix A can be expressed as the sum of a symmetric matrix

and a skew-symmetric matrix.

Hint: Note the identity 퐴 = (퐴 + 퐴 ) + (퐴 − 퐴 ).

The product of symmetric matrices is symmetric if and only if the matrices commute.

Is the product of commuting skew-symmetric matrices skew-symmetric? Explain.

The matrix 퐴 = 0 1−1 0 is skew-symmetric but 퐴퐴 = 퐴 = −1 0

0 −1 is not skew-symmetric. Therefore, the result does not hold.

In general, suppose that 퐴 and 퐵 are commuting skew-symmetric matrices.

Then (퐴퐵) = (퐵퐴) = 퐴 퐵 = (–퐴)(–퐵) = 퐴퐵,

So that 퐴퐵 is symmetric rather than skew symmetric.

Note that if A and B are skew-symmetric and their product is symmetric, then

AB = (AB)T = BT AT = (–B)(–A) = BA,

So the matrices commute and thus skew symmetric matrices, too, commute if and only

if their product is symmetric.

19) Rank of Matrix Def: The number of non zero rows in the row-echelon form of a matrix 퐴 is called

rank of the matrix 퐴. It is denoted as 푟푎푛푘(퐴) or 푟 OR

The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank (A);

The dimension of the null space of A is called the nullity of A and is denoted by

nullity(A).

The reduced row-echelon form of A is

Therefore rank of 퐴 is 2 i.e. 푟푎푛푘(퐴) = 2

Properties:

(a) If 퐴 is any matrix, then 푟푎푛푘 (퐴) = 푟푎푛푘(퐴 ).

Since 푟푎푛푘 (A) = dim(row spaceof A) = dim (column space of A ) = 푟푎푛푘(퐴 )

(b) Dimension Theorem for Matrices

If 퐴 is a matrix with 푛 columns, then 푟푎푛푘(퐴) + 푛푢푙푙푖푡푦(퐴) = 푛

Remark: There are four parameters:

Suppose that 퐴 is an 푚 × 푛 matrix of rank 푟, it follows that 퐴 is an 푛 × 푚 matrix of rank 푟.

Applying dimension theorem for matrices to A and 퐴 yields

푛푢푙푙푖푡푦(퐴) = 푛 − 푟, 푛푢푙푙푖푡푦(퐴 ) = 푚− 푟

Maximum Value for Rank If A is an 푚 × 푛 matrix, then the row vectors lie in ℝ

And the column vectors lie in ℝ .

This implies that the row space of A is at most n-dimensional.

And that the column space is at most m-dimensional.

Since the row and column spaces have the same dimension (the rank of A),

We must conclude that if 푚 ≠ 푛,

Then the rank of A is at most the smaller of the values of m and n.

We denote this by writing 푟푎푛푘(퐴) ≤ min (푚, 푛)

Where min (푚, 푛)denotes the smaller of the numbers m and n if 푚 ≠ 푛

or denotes their common value if 푚 = 푛

Sylvester’s inequality: It states that if 퐴 and 퐵 are 푛 × 푛 matrices with rank 푟 and 푟

respectively, then the rank 푟 of 퐴퐵 satisfies the inequality

푟 + 푟 − 푛 ≤ 푟 ≤ min (푟 , 푟 )

where (푟 , 푟 ) denotes the smaller of 푟 and 푟 or their common value if the two

ranks are the same.

Note that: 1. To confirm this result for some matrices of your choice.

2. The result also true, if 퐴 is 푛 × 푚 matrix and 퐵 푚 × 푝 matrix.

20) System of Linear Equations

An arbitrary system of m linear equations in n unknowns can be written as The following is the system of 3-linear equations and 4-unknown The matrix form of the system is

퐴푋 = 푏 Where A is coefficient 푚 × 푛 matrix, X is variable 푛 × 1 matrix and b is constant 푚 × 1 matrix

Augmented Matrices

If 푏 ≠ 0 then the system 퐴푋 = 푏 is called non homogeneous system of linear equations If 푏 = 0 then the system 퐴푋 = 푏 is called homogeneous system of linear equations