MATLAB Exe School of mathematical ces Xiamen University httplgdipke.xmuedu MATLAB Exercise 3- Linear Space differen of the following are spanning sets for r Justify your answers by at least tw ent methods. (related function may be used rank, rref, det) 1)(0(11 1)10,10 2){0,10,2 2)(2 6 l)lx∈Span(x12x2)? 2)lsy∈Span(x1,x2)? (related function: rref, rank) 3. Which of the following are spanning sets for P3? Justify your answer 1){x2x+ 2)2x-2,x,2x2+ x+1,x-2,x2+3 Help size(A, 1) returns the number of rows ofA size(A, 2 )returns the number of columns ofA 4. Which of the following collections of vectors are linearly independent? 1){1110 2) 3){1-12 5. For each of the sets of vectors in Exercise 3, describe geometrically the span of the given vectors(i.e point out the maximal linearly independent subset for each of the sets of vectors. 6. Given X x 1)Show that x,, x2, x, are linearly dependent 2)Show that x, and x, are linearly independent Ex3-1MATLAB Exercise 3 School of Mathematical Sciences Xiamen University http://gdjpkc.xmu.edu. Ex31 MATLAB Exercise 3 – Linear Space 1. Which of the following are spanning sets for 3 1 R ¥ ? Justify your answers by at least two different methods. (related function may be used: rank, rref, det) 1) 1 0 1 0 , 1 , 0 0 1 1 ÏÊ ˆ Ê ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Ô Ì ˝ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Ô ÓË ¯ Ë ¯ Ë ¯ ˛ 2) 1 0 1 1 0 , 1 , 0 , 2 0 1 1 3 ÏÊ ˆ Ê ˆ Ê ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Á ˜ Ô Ì ˝ Á ˜ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Á ˜ Ô ÓË ¯ Ë ¯ Ë ¯ Ë ¯ ˛ 3) 2 3 2 1 , 2 , 2 2 2 0 ÏÊ ˆ Ê ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Ô Ì ˝ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Ô ÓË- ¯ Ë- ¯ Ë ¯ ˛ 4) 2 2 4 1 , 1 , 2 2 2 4 ÏÊ ˆ Ê- ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Ô Ì - ˝ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Ô ÓË- ¯ Ë ¯ Ë- ¯ ˛ 2. Given 1 2 1 3 2 9 2 , 4 , 6 , 2 3 2 6 5 x x x y Ê - ˆ Ê ˆ Ê ˆ Ê - ˆ Á ˜ Á ˜ Á ˜ Á ˜ = = = = - Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯ Ë ¯ 1) Is Span 1 2 xŒ (x , x ) ? 2) Is Span 1 2 y Œ (x , x ) ? (related function: rref, rank) 3. Which of the following are spanning sets for P3? Justify your answer. 1) { } 2 2 1, x , x +1 2) { } 2 2, x - 2, x, 2x +1 3) { } 2 x +1, x - 2, x + 3 Help size(A,1) returns the number of rows of A size(A,2) returns the number of columns of A 4. Which of the following collections of vectors are linearly independent? 1) 1 0 1 1 , 1 , 0 1 1 1 ÏÊ ˆ Ê ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Ô Ì ˝ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Ô ÓË ¯ Ë ¯ Ë ¯ ˛ 2) 2 2 1 , 2 2 0 ÏÊ ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Ô Ì ˝ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Ô ÓË- ¯ Ë ¯ ˛ 3) 2 2 4 1 , 1 , 2 2 2 4 ÏÊ ˆ Ê- ˆ Ê ˆ ¸ ÔÁ ˜ Á ˜ Á ˜ Ô Ì - ˝ Á ˜ Á ˜ Á ˜ ÔÁ ˜ Á ˜ Á ˜ Ô ÓË- ¯ Ë ¯ Ë- ¯ ˛ 5. For each of the sets of vectors in Exercise 3, describe geometrically the span of the given vectors. (i.e. point out the maximal linearly independent subset for each of the sets of vectors.) 6. Given 1 2 3 1 3 0 2 , 4 , 10 3 2 11 x x x Ê - ˆ Ê ˆ Ê ˆ Á ˜ Á ˜ Á ˜ = = = Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯ 1) Show that 1 2 3 x , x , x are linearly dependent. 2) Show that 1 x and 2 x are linearly independent