Linked Questions

10
votes
3answers
3k views

Will $2$ linear equations with $2$ unknowns always have a solution?

As I am working on a problem with 3 linear equations with 2 unknowns I discover when I use any two of the equations it seems I always find a solution ok. But when I plug it into the third equation ...
6
votes
7answers
8k views

Existence of least squares solution to $Ax=b$

Does a least squares solution to $Ax=b$ always exist?
5
votes
3answers
12k views

Unique least square solutions

There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. But can we find a counter-example to ...
6
votes
3answers
1k views

Why does $A^TAx = A^Tb$ have infinitely many solution algebraically when $A$ has dependent columns?

This is a problem from least square approximation, where we solve the equation $A^TAx = A^Tb$ when $Ax = b$ is unsolvable. The case I am dealing with is when A has dependent columns, i.e. A is an m by ...
3
votes
1answer
2k views

Exact solution of overdetermined linear system

Given a (possibly) overdetermined linear system $Ax=b$, where $A$ is full rank and $A \in \mathbb{R}^{m \times n}, \quad m \ge n$ Does the least squares method provide an exact solution (instead of ...
3
votes
2answers
2k views

Is a least squares solution to $Ax=b$ necessarily unique

Let $A$ be an $m$ x $n$ matrix, and suppose that $b\in\mathbb{R}^n$ is a vector that lies in the column space of $A$. Is a least squares solution to $Ax=b$ necessarily unique? If so, give a detailed ...
3
votes
1answer
1k views

Moore-Penrose Inverse as least-squares solution

I'm trying to understand a conclusion in [1], [2]. There they state a definition and a theorem: Definition(5.2 in [1], 2.2 in [2]): For a general linear System $Ax=y$, we say that $\hat{x}$ is a ...
3
votes
1answer
1k views

Let $A$ be an $8 \times 5$ matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. Show $Ax=b$ must be inconsistent.

Here's the entire question: Let $A$ be an 8 $\times$ 5 matrix of rank 3, and let $b$ be a nonzero vector in $N(A^T)$. a) Show that the system $Ax = b$ must be inconsistent. Gonna take a wild stab at ...
1
vote
1answer
282 views

Query about the Moore Penrose pseudoinverse method

I have recently discovered the Moore-Penrose psuedoinverse method, and I am currently testing the waters with it. I noticed if I have a system, say $$a_1x_1=0$$ $$a_2x_1+a_3x_2=0$$ $$\vdots$$ $$...
1
vote
1answer
192 views

null space of a matrix for a matrix with a 0 column

I'm not a mathematician. Math is a hobby and currently I'm learning bits and pieces of linear algebra when I have time. Confused about a null space for the following matrix \begin{bmatrix} 0 & 0 ...