Independence, Basis and Dimension

1. Independence, Basis and
Dimension
Introduction to
LINEAR ALGEBRA

2. Linear Independence
 What is linear independence?
 Elaboration.
Presented by:-
ATUL KUMAR YADAV (B.TECH computer

3. 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.

4. 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.

5. 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.

6. 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.

7. BASIS
OF
VECTOR SUBSPACE
Introduction to LINEAR ALGEBRA

8. 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.

9. Dimension of a space is the number
of vectors in every basis.
or
Every basis for the space has the same
no. of vectors and this number is dimension.

10. 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).

11. The Dimensions of Four
Fundamental Subspaces
Introduction to LINEAR ALGEBRA

12. Definitions
• Rank: the number of nonzero pivots; the number of independent rows.
• Notation for rank: r
• Dimension: the number of vectors in a basis.

13. The Four Fundamental Subspaces A is
an m x n matrix
Notation Subspace of Dimension
Row Space r
()T RA
Column Space r
R(A)
m 
Nullspace n - r
N(A)
Left Nullspace m - r
( ) T N A
n 
n 
m 

14. The Four Fundamental Subspaces A is an m x n matrix
Description
Row Space
TA
Column space of .
All linear combinations of the columns of .
Column Space All linear combinations of the columns of A.
T A
Nullspace All solutions to Ax = 0.
Left Nullspace All solutions to y = 0.
TA

15. 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.

16. RANK OF MATRIX
Introduction to
LINEAR ALGEBRA

17. ECHELON FORM
 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

18. 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.

19. A = 3 -1 2
-3 1 2
-6 2 4
3 -1 2
0 0 4
R2->R2+R1
0 0 8
R3->R3+2R1
1 -1/3 2/3
R3->1/3R1
0 0 4
0 0 8

20. 1 -1/3 2/3
R2(1/4)
0 0 1
0 0 1
R3(1/8)
1 -1/3 2/3
0 0 1
0 0 1
RANK = No. OF NON ZERO ROW
= 2.

21. THANK YOU
Presented by:-
ATUL KUMAR YADAV