Which solution is better?

[intervade]

I have a homework problem that asks which solution is better while considering the following:

\(\displaystyle
\begin{bmatrix}0.780 & 0.563\\
0.913 & 0.659
\end{bmatrix}
\begin{bmatrix}
x_{1}\\
x_{2}
\end{bmatrix}
=
\begin{bmatrix}
0.217\\
0.254
\end{bmatrix}
\)

Which solution is better?
\(\displaystyle \widehat{x} = (0.341, -0.087) \) or
\(\displaystyle \widetilde{x} = (0.999, -1.001)\).

The answer in the book doesn't make sense to me:
residual vector: \(\displaystyle \widehat{x} = A\widehat{x}-b\), error vector: \(\displaystyle \widehat{x} = \widehat{x}-x\) where x = (1, -1)

Could someone elaborate on this for me? It's to my understanding the the hat represents equations which have the same slope and the same y intercept(duplicate equations) and the tilde represents parallel equations. I'm not sure how this applies to these solutions.

 

[tkhunny]

I wish I understood the question. As it is presented, THE solution is is <1,-1>, so it would seem reasonable that x-tilde would be "better". However, in order to judge "better", one must know the criteria for judging.

What are you doing?
What section are you in?
Waht other kinds of problems are you tackling?

 

[intervade]

I wish I understood the question. As it is presented, THE solution is is <1,-1>, so it would seem reasonable that x-tilde would be "better". However, in order to judge "better", one must know the criteria for judging.

What are you doing?
What section are you in?
Waht other kinds of problems are you tackling?

The chapter basically covers simple row operations and general forms of a matrix. The chapter is titled "Solving systems of linear equations". There were some questions like "use the theory linear equations to solve this simple system.." but other than that there wasn't anything that stood out that could relate to this problem. Maybe I need to solve the system before determining the solution but that doesn't explain the rest of the "answer" (i.e. \(\displaystyle \widehat{x} = A\widehat{x} - b\) , etc..)

 

[tkhunny]

Very odd. Maybe a lesson on rounding error?

 

[SlipEternal]

...
error vector: \(\displaystyle \widehat{x} = \widehat{x}-x\) where x = (1, -1)

I am rather confused by the error vector. If \(\displaystyle \widehat{x} = \widehat{x}-x\) then you are no longer dealing with a vector space, as \(\displaystyle 0=-x=x\ne (1,-1)\).

Going by what is given, you have:

\(\displaystyle \widehat{x} = A\widehat{x} - b\)

\(\displaystyle I\widehat{x} = A\widehat{x} - b\)

\(\displaystyle I\widehat{x} - A\widehat{x} = -b\)

\(\displaystyle (I-A)\widehat{x} = -b\)

\(\displaystyle (A-I)\widehat{x} = b\)

I am not sure what you can do from there. Does that help at all?