Linear Algebra Least Squares Question

In summary, the conversation discusses finding a matrix B that will produce the projection onto the column space of A, even when AT A is not invertible. It is suggested to consider rearranging or deleting dependent columns, but it is also mentioned that this may affect the answer. The use of a basis for the column space is also mentioned as a possible solution.
  • #1
flyingmuskrat
4
0

Homework Statement



Suppose the columns of A are not independent. How could you find a matrix B so that P=B(BTB)^-1BT does give the projection onto the column space of A? (The usual formula will fail when AT A is not invertible).

T is transpose.

Homework Equations


The Attempt at a Solution



I think this is a thought question or something? Do you rearrange the columns...or just delete the dependent columns? But wouldn't that mess up the answer? Idk I got the rest on my p-set but this one I just have nooo idea. I feel like it's really obvious and I'm just missing it. And it's not gram-schmidt or something because that's the section after.
 
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  • #2
how about considering a basis for the column space?
 

1. What is the purpose of least squares regression in linear algebra?

The purpose of least squares regression in linear algebra is to find the line or curve that best fits a set of data points. It is used to model the relationship between two or more variables and make predictions based on that relationship.

2. How is the least squares method used to solve linear algebra problems?

The least squares method is used to minimize the sum of squared residuals between the data points and the predicted values on a line or curve. It involves finding the values of the coefficients that result in the smallest possible error, and those coefficients can then be used to solve the linear algebra problem.

3. What assumptions are made when using least squares regression in linear algebra?

The main assumptions made when using least squares regression in linear algebra are that the data is normally distributed, the relationship between the variables is linear, and there is no significant correlation between the independent variables. Additionally, the residuals should be randomly distributed and have constant variance.

4. Can the least squares method be used for non-linear relationships?

Yes, the least squares method can be used for non-linear relationships by transforming the data or using a non-linear regression model. However, the line or curve would still be the best fit for the transformed data, not the original data.

5. How is the quality of the least squares fit determined?

The quality of the least squares fit is typically determined by the coefficient of determination (R-squared) and the root mean squared error (RMSE). R-squared measures how well the model explains the variation in the data, while RMSE measures the average distance between the data points and the predicted values. A higher R-squared and lower RMSE indicate a better fit.

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