Solving the System Ax = b: Is Full Rank Necessary?

  • Thread starter Thread starter BobSea
  • Start date Start date
  • Tags Tags
    rank System
AI Thread Summary
The discussion centers on whether the system Ax = b has a solution when A is an m x n matrix with m > n and has full rank. It is asserted that the statement is false because vector b does not necessarily lie within the range of A, which would mean Ax = b has no solution. Participants suggest that a counterexample is needed, specifically a simple matrix A composed of standard basis elements in R^m. The challenge lies in constructing a valid example that demonstrates this scenario. Ultimately, the conclusion is that full rank is not a sufficient condition for the existence of a solution to Ax = b.
BobSea
Messages
1
Reaction score
0

Homework Statement


The system Ax=b, with Amxn, and m>n, always has a solution when A has full rank. If False, give a counter example, if True, say why.



Homework Equations


None


The Attempt at a Solution


I want to say False because b doesn't need to be in the range of A, so Ax=b wouldn't have a solution. However, I'm having trouble making a counter example (i.e. A = 3x2 matrix and b = 3x1 matrix) that proves the point.
 
Physics news on Phys.org
Your reasoning is correct. For a counterexample, try a very simple A whose columns are elements of the standard basis for \mathbb{R}^m.
 

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

Similar threads

Left and right inverses of a non-square matrix
Replies
7
Views
3K
Is A Full Rank Equivalent to an Overdetermined System?
Replies
1
Views
2K
A gardener collected 17 apples...
Replies
32
Views
2K
MHB Check the statements about a 4x5 matrix with rank 2.
Replies
9
Views
5K
I A true or false statement concerning condition number of a matrix
Replies
2
Views
659
Matrix-Vector Form Write an Augmented Matrix
Replies
4
Views
3K
The Maximum Rank of a Matrix B Given AB=0 and A is a Full Rank Matrix
Replies
5
Views
2K
Solve Sin(ax-b) Homework: Find Values of a & b
Replies
5
Views
4K
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
2K
The discriminant of a quadratic and real solutions
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top