Testing for Linear Dependence (Matrix Errors when running)

Hi,

What I'm trying to do should be straight forward, I think. First, I have 2 vectors, and I'm testing for linear dependence by A*x = b.

r = [2 1]
s = [3 2]

In MATLAB, I did:

A = [2 3; 1 2]
b = [0; 0]

inv(A) * b

Results shows that x = [0; 0], which is the correct answer.

However, when trying to add a 3rd vector, 't' to the set, things do not seem to work:

r = [2 1]
s = [3 2]
t = [1 2]

So that now;

A = [2 3 1; 1 2 2]

but when I try,

inv(A) * b

I get this error: "??? Error using ==> inv Matrix must be square."

So then I tried this to test it out:

A = [2 3 0; 1 2 0]

and I get this error: ??? Error using ==> mtimes Inner matrix dimensions must agree.

I am not sure what I am doing wrong. Any ideas how to fix this?

Answer 1

[Wayne King]

A rectangular matrix cannot have a true inverse.

How about just

 A = [2 3 0; 1 2 0];
 rref(A)

To get the reduced row echelon form?

From the above you can see that the 3rd column, A(:,3), is -4 times the 1st column plus 3 times the second column

-4*A(:,1)+3*A(:,2)

Of course 3 vectors in R^2 which is what you have in A cannot be a linearly independent set.

Also, just

rank(A)

will tell you the dimension of the range of the columns of A. In this case, rank(A)=2