Computer Project. Purpose: To better understand the notion of rank and learn its connection with linear independence.

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1 MA M. Kon Extra Credit Assignment 3 Due Tuesday, 3/28/17 Computer Project Name Remember - it's never too late to start doing the extra credit computer algebra problem sets! 1. Rank and Linear Independence Purpose: To better understand the notion of rank and learn its connection with linear independence. Prerequisite: Basic knowledge of rank, MATLAB functions used:, rank, rref; and indat, randomint from Lay's Toolbox. Definition. Recall that the rank of a matrix E can be defined to be the number of pivot columns in E. Notice this is well defined since the reduced echelon form of a matrix is unique. One way to find rank is to calculate the reduced echelon form and then count the number of pivot columns. Another quicker way is to use MATLAB's rank function. Ô " # $ % % & ' ( Example 1. Let EœÖ (note you can get Eand other matrices through ' * "# "& Õ " " " " Ø the command indat). Type A, rref(a) to see E and Ô "! " #! " # $ ans œ Ö. There are two pivot columns in the reduced matrix so the!!!! Õ!!!! Ø rank of E is 2. Type rank(a) to see ans œ # at once. 1. Use both of the above methods to find the rank of each of the following four matrices. To get the matrices, type indat if you have not already done so. Then type rref(b), rank(b), etc.

2 " # % " # $! Ô " # $ Ô Ô % & ' Fœ " " " Gœ " # " " HœÖ Õ ( ) *! " # Ø Õ $ ' & " Ø Õ & ( * Ø Ô " " " " % " " #! % ( ' " $ " "! "$ #" Iœ " % # #! #$ &' Ö " & $ $& $) "#' Õ " ' % &' &* #&# Ø Record the reduced echelon form of each matrix, circle each pivot column and record the rank: Rank: 2. (hand) Recall the definition of linear independence in Section 1.6. Let QœÒv " v# v5 Óbe a matrix whose columns are v, " v #,v 5Þ Use this definition and the definition of rank above to explain why the following are logically equivalent (i.e., why each implies the other): (a) The set of vectors Ö v,v,,v " # 5 is linearly independent.

3 (b) The rank of Q is 5. Hint: if vectors are L.I., then what is the dimension of their span? You can use results from class or the text if you wish. 3. Use the method of question 2 to answer the questions below. In each case, write the appropriate matrix, use MATLAB to calculate its rank, and record the rank. (To learn how to store a matrix, see Section 1 of the computer project "Getting Started With MATLAB.") Ú Þ ÝÔ " Ô # Ô $ Ô % # $ % & Example 2: (a) The set ÛÖ ßÖ ßÖ ßÖ ß is not linearly independent. Verify this Ý $ % & ' ÜÕ! Ø Õ" Ø Õ" Ø Õ" Øà by entering an appropriate matrix (call it J ), and then computing rref(f). (b) For the appropriate matrix K, type G, rref(g) to find if the set Ú Þ ÝÔ " Ô # Ô $ Ô $ # $ % & ÛÖ ßÖ ßÖ ß Ö ß is linearly independent. Is it?. Ý $ % & ' ÜÕ! Ø Õ" Ø Õ" Ø Õ" Øà (c) Examine the matrices Fß Gß Hß and I in question 1. For which of these matrices is the set of its columns a linearly independent set? (d) Let v1 œ Ð#ß $ß &ß "Ñß v2 œ Ð"ß "ß #ß *Ñ, and v3 œ Ð$ß %ß!ß!ÑÞ Is the set Ö v,v,v linearly independent? Record the matrix you used and its rank: 4. (MATLAB and hand; use your paper) Recall the transpose of a matrix \ is defined to be X the matrix \ whose columns are formed from the corresponding rows of \. For example,

4 " # $ Ô " % X if \œ, then \ œ # &. % & ' Õ $ ' Ø Do at least 20 experiments to compare the rank of a matrix and the rank of its transpose (you do not need to record all of them). Use many different size matrices. In MATLAB, the single quote creates transpose -- for example, typing X, will create the transpose of X. Start by calculating rank(a), rank(a, ), etc. for all the matrices used above; then do the same for some matrices you create yourself. For example, you could execute lines like X = randomint(3,7), rank(x), rank(x, ) What might we conclude is always true about rank Ð\Ñ and rank Ð\ Ñ, based on your experiments? Is there any basis for believing this from the theorems we have learned? X

5 2. Subspaces Purpose: to deepen your understanding of span, basis and dimension. In particular, to understand what is required for two subspaces of R n, which have the same dimension, to be the same sets. Prerequisite: Understanding of subspaces MATLAB functions that may be useful: rank, rref, diary Remarks: This exercise is a bit more challenging than previous extra credit exercises. You will be given two matrices E and F, each having five rows. Question 1 is easy, but you will need to think about question 2. Once you find a method it will not take long to do the calculations. Observe that Col E and Col F are obviously subspaces of R 5. Directions: Use the matrices E and F given below. Employ MATLAB to do the calculations and attach the results. Explain your methods briefly and why they work. No credit unless your methods and explanation are valid! One way to record your work is to just copy the key calculations by hand. An easier way is to create a diary file of your MATLAB session and print that after you finish all calculations. If you want to create a diary file to record your operations in Matlab, here is a way: start MATLAB and type diary c:subsp before doing any calculations. This will cause everything that appears on the screen after that to be stored in a text file called subsp on your c: drive (of course you can replace c:subsp by any appropriate file name for your system). When your calculations are finished, type diary off (or exit MATLAB) to close the file; then use your favorite editor to print the file subsp. If you want, you can first clean up the file, add titles, etc. before printing it. Problem: Type in (carefully!) the matrices: Ô "!! " " # "!! " Eœ " $ # #! Ö "! "!! Õ # $! " & Ø

6 Ô! "! " "! " " Fœ #! " # Ö! " " " Õ % " $ & Ø 1. Verify that Col E and Col F have the same dimension. 2. Determine whether or not Col E and Col F are the same subspace of R &. Explain what you calculated and why it worked. Notice this is not obvious. For example, if two subspaces of each have dimension 1, each will be a line through the origin, but they might not be the same line. If each has dimension 2, they are planes through the origin, but they might not be the same plane. In general if two subspaces of R 8 have the same dimension 5, we can visualize each as looking like R 5 -- but they might not be exactly the same sets. Your job here is to figure out a way to decide if two subspaces of R 8, which have the same dimension, are actually the same set of points, and apply your method to the subspaces Col E and Col F. Hint: 1. Find bases for Col( E) and Col( F) - recall how this can be done from class. 2. Once you have the bases for the two subspaces, think about how to determine whether they are the same subspace or not. 3. In particular, if you are given a vector, how can you decide whether it is in the span of another collection of vectors? If each vector in the basis for Col( EÑ is a L.C. of vectors in the basis of Col( F), what does this say about the relation of the subspaces Col( E) and Col( F)? An easier way: There is also a very quick way to do this part of the assignment, which you can use if you can figure it out. Think about how you would do this problem if you were working with row spaces instead of column spaces. Try to use this idea in a valid way for your columns. As a hint, the function transpose(a) might be useful - note that this function is entirely equivalent to writing A', as indicated above. R $