Independence, basis and dimension
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This document discusses linear independence, basis, and dimension in linear algebra. It defines linear independence as vectors being linearly independent if the only solution that produces the zero vector is the trivial solution with all coefficients equal to zero. A basis is defined as a set of linearly independent vectors that span the vector space. The dimension of a vector space is the number of vectors in any basis of that space. The dimensions of the four fundamental subspaces (row space, column space, nullspace, and left nullspace) of a matrix are defined in terms of the rank of the matrix.
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