Linear Algebra problem (Least Squares?)

In summary, the problem involves finding the minimum distance between two lines in 3-space represented by the points R = (x,x,x) and S = (y,3y,-1). The equation ATAx = ATb is used to find the minimum distance, but the resulting matrix is incorrect. The equation \nabla ||R-S||^2 = 0 is suggested as an alternative method, but the partial derivatives lead to the same equation. The individual is unsure of where to go from here.
  • #1
Pratha
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0
Linear Algebra problem (Least Squares? - Distance between lines)

Homework Statement



We have two points R = (x,x,x) and S = (y,3y,-1). All we know is that they are on lines somewhere in 3-space and that they don't cross. Need to find an x and y that minimize || R - S ||2

Homework Equations



ATAx = ATb

The Attempt at a Solution



I tried using the equation above, i.e. inverting (ATA) and multiplying both sides with that, but the resulting matrix that I got was a 2x1 matrix of zeros. This is definitely not the right answer. I also tried using (C+D(t)-b)2... for each coord and doing a partial derivative for C and D, but I ended up getting the same equation for both derivatives, which I am sure is not right.

I am very confused and not sure where to go from here.
 
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  • #2
Why don't you minimize it in the usual way, [tex] \nabla ||R-S||^2 = 0 [/tex] ?
 
  • #3
clamtrox said:
Why don't you minimize it in the usual way, [tex] \nabla ||R-S||^2 = 0 [/tex] ?

That is what I tried. At least that's what I think I tried. That was where the (C+D(t)-b)2... etc, was about in my previous post. (C+Dx - y)2 + (C+Dx - 3y)2 + (C+Dx + 1)2.

But, since the t (x) values are all x's, they cancel with the two's after I do the partial derivative w/respect to D, and both derivatives end up the same. Is there something I'm missing, or am I doing something wrong?
 

FAQ: Linear Algebra problem (Least Squares?)

1. What is a linear algebra problem?

A linear algebra problem involves solving a system of linear equations using various mathematical methods, such as Gaussian elimination or matrix operations.

2. What is the least squares method?

The least squares method is a mathematical technique used to find the best fit line or curve for a set of data points. It minimizes the sum of the squared differences between the actual data points and the predicted values on the line or curve.

3. How is least squares used in linear algebra problems?

In linear algebra problems, the least squares method is often used to find the best approximation for a system of equations that does not have an exact solution. It is also used for data fitting and regression analysis.

4. What are the applications of solving linear algebra problems using least squares?

The applications of solving linear algebra problems using least squares are vast and include fields such as statistics, engineering, economics, and social sciences. It is used for data analysis, signal processing, image processing, and more.

5. What are the benefits of using least squares in linear algebra?

The least squares method is a useful tool in linear algebra because it provides a method for finding a solution even when a system of equations is inconsistent or has no exact solution. It also allows for the incorporation of measurement errors in data analysis.

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