Non-zero solution of a homogenous system

  • Thread starter sunny110
  • Start date
  • #1
11
0
Hi,
Could anyone give me a proof for the following theorem?
Theorem : Ax=0 has a non-trivial solution iff det(A)=0

Thanks in advance.
/Sunny
 

Answers and Replies

  • #2
22,129
3,297
You will find the proof in books such as "Linear Algebra" by Lang. I suggest you check out that book.
 
  • Like
Likes 1 person
  • #3
129
10
This is easier to understand if you think about the geometric meaning of a determinant.

A matrix [itex]A[/itex] represents a linear transformation in a given basis. The determinant is a property of the matrix, but it's also a property of the linear transformation it represents, independently of any basis. The determinant is simply the signed volume change of the volume element.

Now we can see what a zero determinant means: it means the transformed volume element is zero. This transformation has squished it down into a lower-dimensional subspace, which means the mapped basis vectors can't be linearly independent. That means there's a nontrivial linear combination of basis vectors which maps to 0. In other words, there must be some [itex]x[/itex] in this basis such that [itex]Ax = 0[/itex].
 
  • Like
Likes 1 person

Related Threads on Non-zero solution of a homogenous system

Replies
7
Views
4K
Replies
5
Views
6K
Replies
2
Views
3K
Replies
3
Views
2K
Replies
2
Views
2K
Replies
2
Views
5K
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
6K
Top