CSIR NET EXAM – LINEAR ALGEBRA NOTES

The notes for CSIR NET Mathematics have been prepared according to the Mathematics exam syllabus.

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Table of Content Vector Space Linear Transformation Matrix and their properties Diagonalization of Matrix Inner Product Space Quadratic Equation

Vector Space :

Let (F,+,.) be a field. The elements of F are called scalars. Let V be a non-empty set whose elements are called vectors. Then V is a vector space over the field F (denoted by V(F)) if

Example:

(i) R is a vector space over R denoted by R(R).

(ii) C is a vector space over R denoted by C(R).

(iii) Every field is vector space over its subfield.

Example: R(Q), C (R)

Vector Subspace :

Subspace :- Let V(F) is a vector space and then we say  W is subspace of V if W is itself form a vector spaces over same field F.

Example :- {0} and V always subspace of V.

Linear Combination

Linear Dependence :

Linear Independence :

Basis of A Vector Space

Definition:

A subset S of a vector space V(F) is said to be a basis of V(F), if

(i) S consists of linearly independent vectors.

(ii) S generates V(F) i.e., L(S)=V i.e., each vector in V is a linear combination of a finite number of elements of S.

Example : V = R³ (R) then b = (1, 0, 0), (0, 1, 0), (0, 0, 1) is a standard basis of V.

Dimension of a vector space :

Let V(F) be a vector space over F and b is a basis of V(F). Then number of elements in b (basis) is a dimension of vector space.

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Linear transformations

Let U(F) and V(F) be two vector spaces over the same field F. Then a mapping T : U → V is called a homomorphism or a linear transformation of U to V if T(au + bv) = aT(u) + bT(v) for all those u,v ∈ U and a,b ∈ F

Example:

Kernal of Linear Transformation :

Let T : V → V’ be a Linear Transformation then K = {x ∈ V | T(x) = 0} is called kernal of T and it is denoted by ker(T)

Range of Linear Transformation :

Let T : V(F) → V’(F) be a linear transformation then R(T) = {T(x) | x ∈ V} is called Range of Linear Transformation.

Matrices and their properties

Definition : A set of mn numbers arranged in t he form of rectangular array consisting of m-rows and n-columns is called an m × n matrix or matrix of order m × n and denoted by A =  [a_ij]_m×n.

Trace of a square matrix :

Sum of all diagonal elements of a square matrix A is known as Trace of A.

Example:

Transpose of a matrix :

If A = [a_ij]_m×n then transpose of A is denoted by A^T is defined as A^T  = [b_ij]_nxm.

Example:

Adjoint of a square matrix :

If A = [a_ij]_nxn then B = [A_ij]_nxn is known as cofactor matrix of A where A_ij is cofactor of a_ij in |A| and transpose of B is known as adjoint of A. It is denoted by Adj A = [A_ij]_nxn.

Inverse of A :

Example :

Eigen value :

Eigen Vector :

Characteristic polynomial :

Let A be square matrix of order n then C_A(x) = |A – xI| is a polynomial of degree n called the characteristic polynomial of A.

Minimal polynomial :

The monic polynomial of lowest degree that annihilates a matrix A is called the minimal polynomial of A. and it is denoted by m(x). Also if f(x) is the minimal polynomial of A, the equation f(x) = 0 is called the minimal equation of the matrix A.

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Diagonalization of Matrix

A matrix is said to be diagonalizable matrix if it is similar to a diagonal matrix. i.e., A is diagonalizable if $ D = diag (d₁, d₂, …., d_n) and a non-singular matrix P such that A = PDP^-1 or P^-1AP = D.

Quadratic form :

Note : Each quadratic form can be written as symmetric matrix.

Example:

Bilinearform

Let f : V x V→ F is called bilinear form. If

(1)       f(u, αv + βw) = αf(u, v) + βf(u, w)

(2)       f(αu + βv, w) = αf(u, w) + βf(v, w)

 

Inner Product Space

A vector space V_F is called an inner product space if there is a function f : V x V→ F satisfying the following axioms for u, v, w ∈ V; α, β ∈ F.

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