Practice Problems - Linear Algebra

Transcription

1 Practice Problems - Linear Algebra Shiro Kuriwaki and Connor Jerzak, Math Prefresher 07 August 7, 07. Dot product uv. Find the dot product of the following vectors u and v a (a) u =, v = b c u v = ()a + ()b + ()c = a + b + c (b) u =, v = 4 u v = ( )4 + () + ( ) = (c) u = 0 9 0, v = u v = (9)6 + ()( 5) =

2 . Multiplication AB. Compute the following matrix products. [ ] [ ] (a) 0 0 [ ] (b) 0 0 (c) (d) [ [ 0 0 ] [ ] [ [ ] 0 0 [ ] ] 0 0 ] Page

3 . Google PageRank. We would like to propose a method that measures the importance of a given website in a Internet network (this problem illustrates the basic idea of Google s Page Rank algorithm ). Suppose the network only consists of four websites, numbered,,, and 4. 4 Each node (circles) is a website. An arrow from circle A to circle B means there is a hyperlink in site A to site B. An arrow with heads on both ends means A links to B and B also links to A We want to compute the values of x = (x, x, x, x 4 ), where x k is some measure of the importance of website k. (a) A simple method to come up with a measure x k is to count the number of k s backlinks, or the number of other websites that link to k. What would the values of x be, by that definition of importance? x = 4 (b) As you might have noticed, one problem with the above count- backlinks method is that it ignores the importance of the website from which those backlinks originate. For example, getting a backlink from should count more towards your importance score than getting a backlink from an obscure blog To overcome this shortcoming, let s now define a k s importance score as a weighted sum of the importance score of its backlinks. Weights are assigned equally (a website plays no favorites among its links): if website j links to n j sites, then it contributes n j of x j to each of them. This definition gives a set of equations. For example, for the first website, we can write x = x + x 4 Bryan, Kurt and Tanya Leise The $ 5,000,000,000 Eigenvector: The Linear Algebra behind Google. SIAM Review, 48:. Page

4 Define the importance scores for x, x, x 4 similarly. x = x + x 4 x = x x = x + x + x 4 x 4 = x + x (c) The four equations above form a linear system, Ax = x. Write out the matrix A. A = (d) Set up the linear system you made in part (b). Ax = x x x x x 4 = x x x x 4 (e) Let x = 9, and then solve for x. The magnitude of one number does not matter for our problem because what matters here is the relative ranking and distance between x,..x 4. Solve for x. This is will be effectively solving for the eigenvector (of eigenvalue ) from a system of linear equations by A. But for now you can solve the equations by usual mulitvariable algebra (by rearranging terms or through reducing the problem to row-echelon form). What does the results tell you about the relative importance of each website? With re-arranging and plugging in equations you will get x = Page 4

5 Solving the system by row-echelon form is also possible by setting up the augmented matrix  = A b. But it might be easier to define another coefficient matrix that shifts all the terms involving x k to one side of the equation, leaving zeroes on the other. And if we did not have the x = restriction we would get vectors proportional to this. This tells us that website is most important, followed by websites, 4, and. Page 5

6 Matrix Products again. Let X = 0 0. Compute the following. (a) X X = (b) X X = (c) X 4 X 4 = (d) X 5 X 5 = Page 6

7 5. rank(a). What is the rank of the following matrices? (a) A = rank(a) =. Its row echelon form is (b) B = 0 rank(b) =. Its row echelon form is Page 7

8 6. Ax = b. Solve the following systems of equations for x, y, and z. To practice solving by elimination, set up each problem as a linear system and apply elementary row operations to transform the augmented matrix in row echelon form. x + y + z = (a) x y + z = y z = (b) Use third row to solve for y in terms of z: y = + z Plug in for y intro row one and two and solve system from here. x =, y =, z = x + y z = 8 x + y z = x 4y + z = 6 Add rows and to solve for y, then add rows and and plug in for y to solve for x, then solve for x. x = 7, y =, z = 5 7. Inverse. Let B = performing elementary row operations.. Calculate its inverse, B by setting up an augmented matrix and B = Page 8

9 8. Keeping track of dimensions. Clogg, Petkova, and Haritou (995) give detailed guidance for deciding between different linear regression models using the same data. In this work they define the matrices X, which is n (p + ) rank p +, and Z, which is n (q + ) rank q +, with p < q. They calculate the matrix A = [X X XZ(Z Z) Z X] ]. Find the dimension and rank of A. Dimension is (p + ) (p + ). The rank of X X is min(p +, n) = p +, and the rank of X Z(Z Z) Z X is min(p +, q +, n) = p + (i.e. matrix multiplication cannot increase rank). So the rank of the difference is p + and the matrix is not invertible unless it is full rank, so A is rank p determinants In their formal study of models of group interaction, Bonacich and Bailey (97) looked at linear and nonlinear systems of equations (their interest was in models that include factors such as free time, psychological compatibility, friendliness, and common interests). One of their conditions for a stable system was that the determinant of the matrix A = r a 0 0 r a 0 r must have a positive determinant for values of r and a. What is the arithmetic relationship that must exist for this to true? (4.5 in Gill) det(a) = r(r 0) a(0 a) + 0(0 r) = r + a > 0 a > r Page 9