MATLAB Exercise # 2 Tutorial & Assignment. Tutorial: Use MATLAB to determine Linear Independence of sets of vectors.
(a.) Create a MATLAB "m-file" ("function" file) that accepts as input a matrix "B" and returns a scalar "d" that equals "1" if the vector if the columns of "B" are Linearly Independent and "0" if the columns of "B" are Linearly Dependent.
Here is such a file.
%This program determines whether a set of column vectors is linearly independent %or linearly dependent. It accepts a Matrix %"B" and returns a scalar "d" which equals "1" if %the columns of "A" are Linearly Independent and "0" if they are %Linearly Dependent. function [d]=Dependence(B) C=rref(B); m=length(diag(B(:,1))); n=length(B(1,:)); if n>m d=0; else s=sum(diag(C)); if n>s d=0; else d=1; end end
Page 1 of 5 (b.) Create a MATLAB "m-file" ("function" file) that has the following features. It contains a 4X2 matrix, "A42" with Linearly Independent columns and two 4x1 vectors. One of those two vectors, namely:"X4" does not lie in the "plane" determined by the column vectors of "A42". The other, namely:"y4" does lie in that plane. The file also contains a 3x2 matrix, "A32", and two 3X1 vectors one of which lies in the plane formed by the columns of "A32" while the other does not. A 2X2 matrix is also contained in the file along with two 2x1 vectors. When the file is "Run" it calls the other m-file named "Dependence" and displays matrices "A42","A32", and "A22" augmented with each of the appropriate given vectors along with a message citing whether the augmented matrix posseses Linearly Independent or Linearly Dependent columns.
This is a file that achieves that functionality.
Page 2 of 5 %This file "calls" the m-file titled "Dependence" and %uses the results of that file to determine if a set %of column vectors formed by augmenting a matrix results %in a set of column vectors that is Linearly Independent or %Linearly Dependent. A42=[1 9;3 11;7 13;5 15]; A32=[1 5;3 2;7 4]; A23=[1 3;5 2]; x4=[6;5;11;17]; y4=[10;14;20;20]; x3=[9;7;11]; y3=[11;7;15]; x2=[-1;1]; y2=[-2;3]; R={}; R{1,1}=[A42 x4]; R{1,2}=[A42 y4]; R{1,3}=[A32 x3]; R{1,4}=[A32 y3]; R{1,5}=[A23 x2]; R{1,6}=[A23 y2]; R{2,1}=['Set is Linearly Independent';'Set is Linearly Dependent ']; for k=1:1:6; fhandle=@Dependence; [d]=Dependence(R{1,k}); if d==0 display(R{1,k}) display(R{2,1}(2,:)) elseif d==1 display(R{1,k}) display(R{2,1}(1,:)) end end
If this file is "Run", here are the results.
Page 3 of 5 ans =
1 9 6 3 11 5 7 13 11 5 15 17
Set is Linearly Independent ans =
1 9 10 3 11 14 7 13 20 5 15 20
Set is Linearly Dependent ans =
1 5 9 3 2 7 7 4 11
Set is Linearly Independent ans =
1 5 11 3 2 7 7 4 15
Set is Linearly Dependent ans =
1 3 -1 5 2 1
Set is Linearly Dependent ans =
1 3 -2 5 2 3
Set is Linearly Dependent Page 4 of 5 Assignment: (a.) Create two m-files using those that I provided above as templates. You will use different matrices, namely: ⎛2 7 4 ⎞ ⎜ ⎟ 3 2 5 ⎛1 4 ⎞ ⎜ ⎟ ⎜ ⎟ ⎛1 −2 ⎞ A53 = ⎜4 5 6 ⎟, A32 = ⎜2 −2 ⎟, and A22 = ⎜ ⎟ and different ⎝1 1 ⎠ ⎜5 6 7 ⎟ ⎜1 3 ⎟ ⎜ ⎟ ⎝ ⎠ ⎝6 7 8 ⎠ ⎛ 6 ⎞ ⎛ 6 ⎞ ⎜ 8 ⎟ ⎜ 8 ⎟ 7 7 ⎜ ⎟ ⎜ ⎟ ⎜⎛ ⎟⎞ ⎜⎛ ⎟⎞ vectors, namely:x5 = ⎜10 ⎟, y5 = ⎜10 ⎟, x3 = ⎜4 ⎟, y3 = ⎜6 ⎟, and ⎜12 ⎟ ⎜12 ⎟ ⎜6 ⎟ ⎜1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ 1 ⎠ ⎝14 ⎠ ⎛3 ⎞ x2 = ⎜ ⎟. Then you will "Run" your program as I did. You will submit ⎝6 ⎠ your 2 m-files along with the results in the "Command" window. (b.) For the Linearly Dependent sets that you found from part (a.), express the given vector as a linear combination of the columns of the corresponding matrix.
Page 5 of 5