Independence, Basis and

Dimension

Introduction to

LINEAR ALGEBRA

Linear Independence

What is linear independence?

Elaboration.

Presented by:-ATUL KUMAR YADAV (B.TECH computer science)

Def. Of Linear Independence

• The column of A are linearly independent when the only solution to Ax = 0 is x=0. No other combination Ax of the columns give the zero vector.

• The sequence of vectors v1,v2…...,vn is linearly independent if the only combination that gives the zero vector is 0v1+0v2+……+0vn.

Def. Of Linear Independence

Let A = { v 1, v 2, …, v r } be a collection of vectors from Rn . If r > 2 and at least

one of the vectors in A can be written as a linear combination of the others,

then A is said to be linearly dependent.

Vector x1,x2,…..,Xn will be independent if no combination gives zero

vector.{except zero combination i.e Ci =0}

C1x1+C2x2+C3x3…….Cnxn != 0.

If one vector is zero the independence is failed.

Examples:-

The vectors (1,0) and (0,1) are independent.

The vectors (1,0) and (1,0.00001) are independent.

The vectors (1,1) and (-1,-1) are dependent.

The vectors (1,1) and (0,0) are dependent because of the zero vaector.

In R^2 , any three vectors (a,b) (c,d) and (e.f) are dependent.

Vectors that Span a Subspace.

Def:- A set of vectors span a space if their linear combinations fills the space.

Vectors v1,….,vn span a subspace means: Space consits of all comb of those

vectors.

The row space of a matrix is the subspace of R^n spanned by the rows.

The row space of A is C(A^T).It is the column space of A^t.

BASIS

OF

VECTOR SUBSPACE

Introduction to LINEAR ALGEBRA

A Basis for a vector space.

Def:- A basis for a vector space is a sequence of vectors with two properties:

The basis vectors are linearly independent and they span the space.

The vector v1,…….vn are a basis for R^n exactly when they are the columns of

an n by n invertible matrix. Thus R^n has infinitely many different bases.

The pivot columns of A are a basis for its column space.

The pivot rows for its row space. So are the pivot rows of its echelon form.

Every basis for the space has the same

no. of vectors and this number is dimension.

Dimension of a space is the number of vectors in every basis.

or

Dimension of C(A)

For Example:-

Rank of Matrix = 2 ,then no. of pivots column is 2 and this is the dimension of

C(A) = 2.

Dimension of Null Space is equals to no. of free variables. { n-r }.

n-r = dimension of N(A).

The Dimensions of Four

Fundamental Subspaces

Introduction to LINEAR ALGEBRA

Definitions

• Rank: the number of nonzero pivots; the number of independent rows.

• Notation for rank: r

• Dimension: the number of vectors in a basis.

The Four Fundamental Subspaces A is an m x n matrix

Notation Subspace of Dimension

Row Space r

Column Space r

Nullspace n - r

Left Nullspace m - r

( )TR A

( )R A

( )N A

( )TN A

n

n

m

m

The Four Fundamental Subspaces A is an m x n matrix

Description

Row SpaceColumn space of .

All linear combinations of the columns of .

Column Space All linear combinations of the columns of A.

Nullspace All solutions to Ax = 0.

Left Nullspace All solutions to y = 0.

TA

TA

TA

Some Notes

The row space and the column space have the same dimension, r.

The row space is orthogonal to the null space.

The column space is orthogonal to the left null space.

RANK OF MATRIX

Introduction to

LINEAR ALGEBRA

FIRST NON-ZERO ELEMENT IN EACH ROW IS 1.

EVERY NON-ZERO ROW IN A PRECEDES EVERY ZERO ROW.

THE NO. OF ZERO BEFORE THE FIRST NON-ZERO ELEMENT IN 1ST,2ND,3RD,……ROW SHOULD

BE INCREASING ORDER.

EX-

1 2 3 1 2 3 4

0 1 4 0 1 2 3

0 0 1 0 0 1 9

0 0 0 1

ECHELON FORM

RANK MATRIX (r)

• IT HAS ATLEAST MINORS OF ORDER r IS DIFFERENT FROM ZERO.

• ALL MINORS OF A OF ORDER HIGHER THAN r ARE ZERO.

• THE RANK OF A IS DENOTED BY r(A).

• THE RANK OF A ZERO MATRIX IS ZERO AND THE RANK OF AN IDENTITY MATRIX OF ORDER n IS n.

• THE RANK OF MATRIX IN ECHELON FORM IS EQUAL TO THE NO. OF NON-ZERO ROWS OF THE MATRIX.

• THE RANK OF NON-SINGULAR MATRIX OF ORDER n IS n.

A = 3 -1 2-3 1 2-6 2 4

3 -1 2 0 0 4

R2->R2+R10 0 8

R3->R3+2R1

1 -1/3 2/3 R3->1/3R1

0 0 40 0 8

1 -1/3 2/3 R2(1/4)

0 0 1 0 0 1

R3(1/8)

1 -1/3 2/3 0 0 10 0 1

RANK = No. OF NON ZERO ROW = 2.

THANK YOU

Presented by:-

ATUL KUMAR YADAV