ATLAB Lecture 3 School of Mathematical Sciences Xiamen University http∥gdjpkc.xmu.edu.cr t If the only way the linear combination k, x,+k,x,+.+k, can equal the zero vector is for all scalars k,, k,,,k, to be 0, then x, x 2,,, are linearly independent The vectors x,,x,,,x, in a vector space V are said to be linearly dependent if there exist scalars k, k2, ..,k, not all zero such that tk 2 If there are nontrivial choices of scalars for which the linear combination K,x+k,x,+.+,r, equals the zero vector, then x,, x,,,x, are linearly dependent A vector x is said to can be written as a linear combination of x,,x,, ,, if there exist scalars k,k2,k, such that x=k,x,+kx2+.+k,x The vectors x,, x,,,x, in a vector space V are said to be a basis of v if x,,x,,,x are linearly independent and for any vector xE V can be written(uniquely)as a linear bination of x,, x ,, xn,. The n is called as the dimension of v, and will be denoted dim(v) Let s denotes a set of vectors xi,x2,,, in a vector space V. x,,x,,,xi is a subset of S If for any xES, x can be written uniquely as a linear combination of xi, i,,x, x,,x,,,x will be called as one of the maximal linearly independent subsets of s Y MATLAB suppose that V=R near relation >>A=fix(10*rand(3, 3)): %Obtain a set of vectors which are the columns of A >>r=rank(A) >> ifr disp"The columns of A are linear disp The columns of A are linearly dependent end linear combinationMATLAB Lecture 3  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu.cn  Lec3­2  ² If the only way the linear combination  1 1 2 2  ... n n  k x + k x + + k x can equal the zero vector is for all scalars  1 2  , ,..., n  k k k to be 0, then  1 2  , ,..., n  x x x are linearly independent.  ¸ The vectors  1 2  , ,..., n  x x x in a vector space V are said to be linearly  dependent if there exist scalars  1 2  , ,..., n  k k k not all zero such that  1 1 2 2  ... n n  k x + k x + + k x .  ² If there are nontrivial  choices of scalars  for which  the linear combination  1 1 2 2  ... n n  k x + k x + + k x equals the zero  vector,  then  1 2  , ,..., n  x x x are linearly  dependent.  ¸ A vector x is said to can be written as  a  linear combination of 1 2  , ,..., n  x x x if there exist scalars  1 2  , ,..., n  k k k such that  1 1 2 2  ... n n  x = k x + k x + + k x .  ¸ The vectors  1 2  , ,..., n  x x x in a vector space V are said to be a basis of V if  1 2  , ,..., n  x x x are linearly independent  and for any vector x ŒV can be written (uniquely) as  a linear combination of 1 2  , ,..., n  x x x . The n is called as the dimension of V, and will be denoted by  dim(V) .  ¸ Let S denotes a set of vectors  1 2  , ,..., n  x x x in a vector space V.  1 2  , ,..., r  i i i  x x x is a subset of S. If for any  x ŒS, x can be written uniquely as a linear combination of 1 2  , ,..., r  i i i  x x x ,  1 2  , ,..., r  i i i  x x x will be called as one of the maximal linearly independent subsets of S.  ² MATLAB suppose that  1 V R n¥ = linear relation  >> A = fix (10*rand(3, 3));  % Obtain a set of vectors which are the columns of A  >> r = rank(A);  >> if r == 3  disp ‘The columns of A are linearly independent.’  else disp ‘The columns of A are linearly dependent.’  end  linear combination