Linear Algebra

Linear Algebra

Chapter 3

Question 1

If the above data points actually lay on a straight line y = C+Dt, we have:

 =

Call the matrix A and the vector on the right-hand side. Of course this system is inconsistent, but we want to find  such that  is as close as possible to . As we’ve seen, the correct choice of  is given by:

To compute this, first note that

Therefore,

And so

Therefore, the best-fit line for the data is:

Here are the data points and the best-fit line on the same graph:

Question 13

Let  Then, we set  and

Question 14

We now compute If we denote then by the Gram-Schmidt process,

Question 16

Chapter 4

Question 5

Question 6

Let be any basis of a -dimensional subspace of Then by Gram-Schmidt orthogonalisation process, we get an orthonormal set with and for

Question 7

Question 16

Question 17

Question 24     

Question 25

Suppose we have matrix of dimension with Then by the application of the Gram-Schmidt orthogonalisation process yields a set of orthonormal vectors of In this case, for each we have

Question 14

Question 3

References

Bretscher, O. (2004). Linear Algebra with Applications, (3^rd ed.).  New York, NY: Prentice Hall.

Farin, G., & Hansford, D. (2004). Practical Linear Algebra: A Geometry Toolbox. London: AK Peters.

Friedberg, S. H., Insel, A. J., & Spence, L. E. (2002). Linear Algebra, (4^th ed.). New York, NY: Prentice Hall.

Leon, S. J. (2006). Linear Algebra with Applications, (7^th ed.). New York, NY: Pearson Prentice Hall.

McMahon, D. (2005). Linear Algebra Demystified. New York, NY: McGraw–Hill Professional.

Zhang, F. (2009). Linear Algebra: Challenging Problems for Students. Baltimore, MA: The Johns Hopkins University Press.