厦门大学数学科学学院：《高等代数》课程教学资源（应用与实验）MATLAB Ex 3 - Linear Space（10）

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 Ex3-1

MATLAB 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

MATLAB Exercise 3 School of Mathematical Sciences Xiamen University httplgdipke.xmuedu 3)What is the dimension of Span(X,,x2, x3)? 4)Give a geometric description of Span(Xi,x2,x,) 7. The vectors 10 7 11 Span R. Pare down the set X,, x2,x,, x4, x, to form a basis for R 8.Lex=(2 Find the coordinates(Akt)of x with respect to,,x (i.e. A=(x1,x2), the coordinates of x with respect to,, x, is c=A x) 9. Let x x be a basis of r. Find the transition matrix from the standard basis er e2) to the basis ( x1, x2) and determine the coordinates ofx under the basis(x1, x2i 7 10. Find the transition matrix corresponding to the change of basis from x, x2]toy,y2I 3 where x=l,x2 y 11. For each of the following matrices, find a basis for the row space (iJ/a]), a basis for the column space(列空间), and a basis for the null space(零空间={x|Ax=0}) of A. The subspace of rx by the column vectors of A is called the column space ot Mac (If A is an m Xnmatrix, the subspace of r by the row vectors of A is called the row sp A =-2 36 2)A=3256 3)A=3256 12. *Generate 10X 10, 100 X 100, 1000 X 1000 random matrices, and corresponding column vectors. Compare the time used in solving linear system Ax = b by different functions or operators such as inv, \ lu and qr.(Refer to Lecture 3) Help Select Matlab Help in the toolbar, then select Index and input gr( and lu )to see different usage of these functions, understand what meaning of [L, U, P]= lu(A,0.0); [L, U, P]=lu(A, 1.0) Q, RI=qr(A, 0) tic, operation, toc prints the number of seconds required for the operation Ex3-2

MATLAB Exercise 3  School of Mathematical Sciences Xiamen University  http://gdjpkc.xmu.edu. Ex3­2  3) What is the dimension of Span ( 1 2 3  x , x , x )? 4) Give a geometric description of Span ( 1 2 3  x , x , x ).  7.  The vectors  1 2 3 4 5  1 3 0 2 1 2 , 4 , 10 , 7 , 3 3 2 11 3 2 x x x x x Ê - ˆ Ê ˆ Ê ˆ Ê ˆ Ê - ˆ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ = = = = = Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Á ˜ Ë ¯ Ë ¯ Ë ¯ Ë ¯ Ë ¯ Span R 3 . Pare down the set  1 2 3 4 5  x , x , x , x , x to form a basis for R3 .  8.  Let  1 2  3 1 10  , , 2 3 7 x x x Ê ˆ Ê ˆ Ê ˆ = Á ˜ = Á ˜ = Á ˜ Ë ¯ Ë ¯ Ë ¯ .  Find the coordinates (坐标) of x with respect  to  1 2  x , x .  (i.e. A=(x1,x2), the coordinates of x with respect to  1 2  x , x is c=A ­1 x).  9.  Let  1 2  2 1  , 4 3 x x Ê ˆ Ê ˆ = Á ˜ = Á ˜ Ë ¯ Ë ¯ be a basis of R2 . Find the transition matrix from the standard basis {e1, e2} to the basis {x1, x2} and determine the coordinates of 10  7 x Ê ˆ = Á ˜ Ë ¯ under the basis {x1, x2}.  10.  Find the transition matrix  corresponding  to the change of basis  from  1 2  [x , x ]to  1 2  [y , y ],  where 1 2  2 1  , 4 3 x x Ê ˆ Ê ˆ = Á ˜ = Á ˜ Ë ¯ Ë ¯ ,  1 2  1 3  , 2 4 y y Ê- ˆ Ê ˆ = Á ˜ = Á ˜ Ë ¯ Ë ¯ .  (X ­1 Y) 11.  For each of the following matrices, find a basis for the row space（行空间）, a basis for the column space（列空间）, and a basis for the null space (零空间={x | Ax = 0}).  （If A is an m×n matrix, the subspace of R1×n by the row vectors of A is called the row space of A. The subspace of Rm×1 by the column vectors of A is called the column space of A.） 1) 1 2 3 2 3 6 9 4 6 A Ê ˆ Á ˜ = - Á ˜ Á ˜ Ë ¯ 2) 1 2 1 0 3 2 5 6 2 2 0 4 A Ê - ˆ Á ˜ = Á ˜ Á ˜ - Ë ¯ 3) 1 2 1 0 3 2 5 6 2 0 6 6 A Ê - ˆ Á ˜ = Á ˜ Á ˜ Ë ¯ 12.  *Generate 10×10,  100×100,  1000×1000  random  matrices,  and  corresponding  column  vectors.  Compare the time used in solving linear system Ax = b by different  functions  or operators such as inv, \, lu and qr. (Refer to Lecture 3) Help Select Matlab Help in the toolbar, then select Index and input qr ( and lu )to see different  usage of these functions,  understand what meaning of [L,U,P] = lu(A,0.0); [L,U,P] = lu(A,1.0);  [Q,R] = qr(A,0) .  tic, operation, toc prints the number of seconds required for the operation