MATH APPLIED MATRIX THEORY


 Clyde Shields
 1 years ago
 Views:
Transcription
1 MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points each:. Draw the web and its baclins. Solution: This is easy to do: just remember how a matrix describes a directed graph. For instance, since the element A(, 3 equals, then there is an arrow (baclin going from page to page 3. Since A(4,, then there is no baclin going from page 4 to page, and so on. 2. Find the importance score matrix B for the web. Solution: Once the web is understood, the importance scores x,..., x 5 verify the following x 3 x x 5 x 2 2 x + 3 x x 5 x 3 2 x In matrix notation, we obtain x x 2 x 3 x 4 x 5 x 4 x x 3 x 5 x 4 /3 /2 /2 /3 /2 /2 /3 x x 2 x 3 x 4 x 5
2 Hence, B /3 /2 /2 /3 /2 /2 /3 Notice that all the columns of B add up to (that is, B is column stochastic. This is a quic way to chec that B is not incorrect. 3. Let M.85 B +.5 /5 ones(5 and suppose that MATLAB gives >> [P D eig(m P i i i i i i i i D i i.482 Find the importance score vector x [x, x 2, x 3, x 4, x 5. Solution: The importance score vector must verify x Mx and x +x 2 +x 3 +x 4 +x 5. In particular, the equation x Mx says that x must be an eigenvector of M with eigenvalue λ. In the matrix D the eigenvalue λ is in the first column, so that one associated eigenvector will be the first column of P. That is, the vector v [.352,.499,.23,.5488,.538. Clearly, the sum of the components of this eigenvector v is not equal to. To create an eigenvector whose components add up to, we divide the vector v by the sum of its components. That is, x [.352,.499,.23,.5488, [.352,.499,.23,.5488, [.635,.233,.995,.2562,.2478 Hence the score importance vector is x [.635,.233,.995,.2562,
3 PROBLEM 2 (25 points Determine whether the following matrices are linearly independent M [ 2 (This is problem 2, section 7.2 [, M 2 [ 5 3, M [ 2, M 4 3 Solution: By definition, M, M 2, M 3, M 4 are linearly independent if whenever there are real numbers x, x 2, x 3, and x 4 such that x M + x 2 M 2 + x 3 M 3 + x 4 M 4, then necessarily x x 2 x 3 x 4. The equation x M + x 2 M 2 + x 3 M 3 + x 4 M 4 means ( x M + x 2 M 2 + x 3 M 3 + x 4 M 4. That is, x [ 2 + x 2 [ + x 3 [ x 4 [ 2 3 ( Thus, componentwise this implies That is, we end up with the system If we denote A x + x 2 + 5x 3 + 2x 4 x 2 + 3x 3 + x 4 x 2 3x 3 x 4 2x + x 2 + 5x 3 + 3x x x 2 x 3 x 4, then x is a solution to the system Ax. Linearly independence of M, M 2, M 3, and M 4 means that there is exactly one solution to Ax, that is, the zero solution. Let s see if that s the case. By computing the reduced row echelon form of the augmented matrix [A b, where b [,,,, (which is very simple to do by hand, we get that there is one free variable. Therefore, there are infinitely many solutions. This is telling us that M, M 2, M 3, and M 4 are linearly dependent since there are infinitely many nonzero numbers x, x 2, x 3, and x 4 such that x M + x 2 M 2 + x 3 M 3 + x 4 M 4. 3
4 PROBLEM 3 (3 points Determine whether the following sets are vector spaces ( points each. U {a + b t such that a, b R} (This is problem 28, section 7. Solution: By definition. U is a vector space if whenever the sum of two objects in U is also an object in U, and whenever an object is U is multiplied by a real number the result is also an object in U. To chec the first condition let s tae two objects u and v in U. By the definition of U u and v must be of the form u + l t, v r + s t where, l, r and s are some fixed real numbers (here t is the variable. Now, the sum u + v is u + v ( + r + (l + s t, which is also of the form real number + real number t. Hence, u + v is also an object of U. Now tae a real number λ and an object u U. Then, u must be of the form u m + y t for some real numbers m and y. Therefore, the product λu is given by λu λ(m + y t λm + λy t, which is also of the form real number + real number t. Hence, λu is also an object of U. Therefore, U is a vector space. {[ } a b 2. V such that a + d (This is problem 2, section 7. c d Solution: By definition, the objects of V are those 2 2 matrices whose diagonal elements add up to. Given two objects u and v in V, they must be of the form [ [ e f i j u, v g h l for some real numbers e, f, g, h, i, j, and l such that e + h and i + l. The sum u + v is then [ e + i f + j u + v g + h + l We only need to chec whether the diagonal elements of u + v add up to. The diagonal elements of u + v are e + i and h + l. Hence That, is u + v is also an object in V. (e + i + (h + l e + h + i + l +. Now, tae a real number λ and an object u V. Therefore, since u must be of the form [ p q u r s for some real numbers p, q, r, and s with p + s. The product λu is given by [ λp λq λu λr λs 4
5 It remains to chec whether the diagonal elements of λu add up to zero. We have λp + λs λ(p + s λ. Hence, λu is also an object of V, for every real number λ and every object u in V. Thus, V is a vector space. 3. The set S consisting of all 2 2 symmetric matrices. (This is problem 23, section 7. Solution: Given any two objects u and v of S, they must be of the form [ [ p q a b u, v q s b c for some real numbers p, q, s, a, b, and c. Notice that u and v are symmetric, that is, u u T and v v T (they equal their transposes. Now, u + v is given by [ p + a q + b u + v q + b s + c Notice that u + v is also symmetric. Therefore, u + v is also an object in S. Now tae a real number λ and an object u S. u must be of the form [ r t u t s for some real numbers r, s, and t. Hence, the product λu is given by [ λr λt λu, λt λs which is also a symmetric matrix. Hence, λu is also an object in S. Therefore, S is a vector space. 5
6 PROBLEM 4 (25 points The system of difference equations { x x 2y y 2x + y where, 2, 3,... defines a discrete dynamical system. Find a formula for u [x, y for every natural number using that x y and that MATLAB gives >> [P Deig([ 2; 2 (This is problem 2, section 6.3 [ / 2 / 2 P / 2 / 2 [ D 3 Solution: Set u [x, y. Clearly, the system above is equivalent to u Au, where the matrix A is given by A [ 2 2 Iterating the equation u Au we get that u A u (where u [x, y [,. On the other hand, by using the MATLAB information we see that A is diagonalizable (here we use the quic chec that its eigenvalues are all different. In particular, we have which gives [ A P D P P 3 A P D P, [ P ( P 3 P. [ a b Now remember that the inverse of any 2 2 matrix is given by the matrix c d ad bc [ d b c a Using this little tric, we easily compute the inverse of P. P ( / 2 (/ 2 ( / 2 ( / 2 [ / 2 / 2 / 2 / 2 [ / 2 / 2 / 2 / 2 as which gives that P P (that is, in this particular case P turned out to be an orthogonal matrix. All this gives [ A ( P 3 P. 6,
7 Hence, [ u A ( u P 3 Now, we compute the product [ / 2 / 2 / 2 / 2 [ / 2 / 2 P u / 2 / 2 [ ( 3 [ ( 3 [ ( / 2 / 2 / 2 / 2 [ / 2 / 2 [ [ / 2 / 2 / 2 / ( 2 3 / 2 + / 2 [ [ [ / 2 / 2 / 2 / ( 2/ [ [ / 2 / 2 / 2 / ( ( 2/ 2 2 [ ( / 2( ( 2/ 2 ( / 2( ( 2/ 2 [ ( ( [ / 2 / 2 / 2 / 2 [ ( Hence, the general formula for u is u (. That is y ( and x (. In particular, we have that x y, x 2 y 2, x 3 y 3, and so on. ( 7
8 [ PROBLEM 5 (2 points Determine whether the matrix P 2 matrix for A [ 4 2 is a diagonalizing. (This is problem 8 in the Review for Chapter 6, page 323 Solution: By definition, P is a diagonalizing matrix for A if the we have P A P D where D is a diagonal matrix. Hence, we only need to compute the product P A P and chec whether or not it is a diagonal matrix. [ a b To find P we use that the inverse of any 2 2 matrix is given by the matrix c d ad bc In our case, the matrix in question is P P [ d b c a [ 2 [ Hence, its inverse is given by [ 2 Now, we compute [ P A P 2 [ 2 [ 2 3 [ 4 2 [ [ 2 Hence, P A P is a diagonal matrix and therefore P is indeed a diagonalizing matrix for A. 8
9 PROBLEM 6 (2 points Suppose that the characteristic polynomial of a 3 3 matrix B is Answer the following (5 points each:. What are the eigenvalues of B? p(λ (λ (λ + (λ 4. Solution: By definition, the eigenvalues of B are the roots of its characteristic polynomial. That is, the eigenvalue of B are the solutions to Clearly, the eigenvalues are,, and Is B diagonalizable? p(λ (λ (λ + (λ 4 Solution: Yes, it is. The eigenvalues are all distinct (they do not repeat. This is a sufficient (although not necessary condition for diagonalization. 3. What is det(b? Solution: Since B is diagonalizable, we now that B P D P for some invertible matrix P and some diagonal matrix D whose diagonal elements are the eigenvalues of A. Therefore we have det(b det(p D P det(p det(d det(p det(p det(d det(p Thus, det(b Is B invertible? det(d ( 4 4. Solution: Yes, since det(b 4 it follows that B is invertible. 9
10 PROBLEM 7 (25 points Let u [ 3, 2, 4, v [ 2,, and w [, 5,. Compute u (v w. (This is problem 56, section 5.2 Solution: Let s recall the definitions of the dot and cross products (which is a number and [x, x 2, x 3 [y, y 2, y 3 x y + x 2 y 2 + x 3 y 3, [x, x 2, x 3 [y, y 2, y 3 [x 2 y 3 x 3 y 2, x 3 y x y 3, x y 2 x 2 y (which is a vector. We then have v w [ 2,, [, 5, [4, 2, and u (v w [ 3, 2, 4 [4, 2, PROBLEM 8 (25 points Find the shortest distance from the point P ( 2, 3, to the subspace U span{u, u 2 } where u [2,, and u 2 [7, 4, 2. (This is problem 7, section 4.3 Solution: Let v [ 2, 3,. Since u and u 2 are orthogonal (their dot product is zero the projection of v onto the subspace U is given by v v u u 2 u + v u 2 u 2 2 u 2 Recall that the norm of a vector x [x, x 2, x 3 is given by x x 2 + x2 2 + x2 3. Computing and plugging in the dot products and the norms in the formula for v we obtain v 7 5 u u [2,, [7, 4, 2. Finally, the distance from P to the subspace U is given by v v
11 PROBLEM 9 (3 points Let T : R 3 R 2 be the linear operator given by T (x, x 2, x 3 (2x 3x 2 + x 3, x + x 3. Determine the following ( points each (This is problem 7, section 3.4:. The matrix that represents T. Solution: The canonical basis for R 3 is B {[,,, [,,, [,, } Hence, the matrix A that represents T is given by A [T (,,, T (,,, T (,, We now use the definition of T to compute ( T (,, + and Hence, 2. What is ran(t? Is T onto? ( T (,, + ( T (,, + A ( 2 3,,,. Solution: ran(t equals col(a. Hence, since a basis for col(a is {( ( } 2 3, (the third column is a linear combination of the first two. We obtain that {( ( } 2 3 col(a span, In particular, ran(a 2, which coincides with the dimension of the target space R 2. Therefore, T is onto. 3. What is er(t? Is T onetoone? Solution: Since er(t null(a and, by definition, null(a consists of those vectors x such that Ax, we obtain that er(t is the subspace formed by the solutions to Ax. This leads to the system ( 2 3 x x 2 x 3 (.
12 Computing the reduced row echelon form of the augmented matrix ( [A [, 2 3, which can be easily done by hand, we obtain that there is one free variable and that er(t turns out to be er(t span ( [3, 3. By definition, T is onetoone only when er(t consists of just the zero vector. In our case er(t consists of infinitely many vectors (all the multiples of [3, 3. Hence, T is not onetoone. PROBLEM (2 points Find the value of such that the angle between the vectors [2, 3, and [ 2, 5, equals π/2. (This is problem 7, section 4. Solution: The vectors forming an angle of π/2 means that they are orthogonal to each other. That is, their dot product has to be zero. Hence, we need to find the value of such that Clearly the answer is 9. [2, 3, [ 2, 5,
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationMath 22 Final Exam 1
Math 22 Final Exam. (36 points) Determine if the following statements are true or false. In each case give either a short justification or example (as appropriate) to justify your conclusion. T F (a) If
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationReview Session There will be a review session on Sunday March 14 (Pi day!!) from 3:305:20 in Center 119.
Announcements Review Session There will be a review session on Sunday March 14 (Pi day!!) from 3:305:20 in Center 119. Additional Office Hours I will have office hours Monday from 23 (normal time) and
More informationMAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
More information(Practice)Exam in Linear Algebra
(Practice)Exam in Linear Algebra First Year at The Faculties of Engineering and Science and of Health This test has 9 pages and 15 problems. In twosided print. It is allowed to use books, notes, photocopies
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationMath 308 Final Exam Winter 2015, Form Bonus. of (10) 135
Math 308 Final Exam Winter 015, 318015 Your Name Your Signature Student ID # Points 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. Form Bonus of 50 13 1 17 8 3 7 6 3 4 6 6 (10) 135 No books are allowed. But you are
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationSection 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =
Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and
More information1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationMath 54 Midterm 2, Fall 2015
Math 54 Midterm 2, Fall 2015 Name (Last, First): Student ID: GSI/Section: This is a closed book exam, no notes or calculators allowed. It consists of 7 problems, each worth 10 points. The lowest problem
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationMA 52 May 9, Final Review
MA 5 May 9, 6 Final Review This packet contains review problems for the whole course, including all the problems from the previous reviews. We also suggest below problems from the textbook for chapters
More informationLinear Algebra A Summary
Linear Algebra A Summary Definition: A real vector space is a set V that is provided with an addition and a multiplication such that (a) u V and v V u + v V, (1) u + v = v + u for all u V en v V, (2) u
More informationSolutions to Assignment 9
Solutions to Assignment 9 Math 7, Fall 5.. Construct an example of a matrix with only one distinct eigenvalue. [ ] a b We know that if A then the eigenvalues of A are the roots of the characteristic equation
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationMATH 2030: EIGENVALUES AND EIGENVECTORS
MATH 200: EIGENVALUES AND EIGENVECTORS Eigenvalues and Eigenvectors of n n matrices With the formula for the determinant of a n n matrix, we can extend our discussion on the eigenvalues and eigenvectors
More informationSOLUTIONS TO PROBLEM SET 6
SOLUTIONS TO PROBLEM SET 6 18.6 SPRING 16 Note the difference of conventions: these solutions adopt that the characteristic polynomial of a matrix A is det A xi while the lectures adopt the convention
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationExamination paper for TMA4115 Matematikk 3
Department of Mathematical Sciences Examination paper for TMA45 Matematikk 3 Academic contact during examination: Antoine Julien a, Alexander Schmeding b, Gereon Quick c Phone: a 73 59 77 82, b 40 53 99
More informationc 1 v 1 + c 2 v c k v k
Definition: A vector space V is a nonempty set of objects, called vectors, on which the operations addition and scalar multiplication have been defined. The operations are subject to ten axioms: For any
More information(a) Compute the dimension of the kernel of T and a basis for the kernel. The kernel of T is the nullspace of A, so we row reduce A to find
Scores Name, Section # #2 #3 #4 #5 #6 #7 #8 Midterm 2 Math 27W, Linear Algebra Directions. You have 0 minutes to complete the following 8 problems. A complete answer will always include some kind of work
More informationLinear Algebra Test File Spring Test #1
Linear Algebra Test File Spring 2015 Test #1 For problems 13, consider the following system of equations. Do not use your calculator. x + y  2z = 0 3x + 2y + 4z = 10 2x + y + 6z = 10 1.) Solve the system
More informationAdvanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur
Advanced Engineering Mathematics Prof. Pratima Panigrahi Department of Mathematics Indian Institute of Technology, Kharagpur Lecture No. # 06 Method to Find Eigenvalues and Eigenvectors Diagonalization
More informationNote: A typo was corrected in the statement of computational problem #19.
Note: A typo was corrected in the statement of computational problem #19. 1 True/False Examples True or false: Answers in blue. Justification is given unless the result is a direct statement of a theorem
More informationMAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
More informationMath 24 Winter 2010 Wednesday, February 24
(.) TRUE or FALSE? Math 4 Winter Wednesday, February 4 (a.) Every linear operator on an ndimensional vector space has n distinct eigenvalues. FALSE. There are linear operators with no eigenvalues, and
More informationSymmetric Matrices and Quadratic Forms
7 Symmetric Matrices and Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES SYMMERIC MARIX A symmetric matrix is a matrix A such that. A A Such a matrix is necessarily square. Its main diagonal entries
More informationMA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam
MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am  :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.
More informationSolutions to Assignment 12
Solutions to Assignment Math 7, Fall 6.7. Let P have the inner product given by evaluation at,,, and. Let p =, p = t and q = /(t 5). Find the best approximation to p(t) = t by polynomials in Span{p, p,
More informationMATH 304 Linear Algebra Lecture 11: Basis and dimension.
MATH 304 Linear Algebra Lecture 11: Basis and dimension. Linear independence Definition. Let V be a vector space. Vectors v 1,v 2,...,v k V are called linearly dependent if they satisfy a relation r 1
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationMATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial.
MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial. Eigenvalues and eigenvectors of a matrix Definition. Let A be an n n matrix. A number λ R is called
More information18.03 LA.4: Inverses and Determinants
8.3 LA.4: Inverses and Determinants [] Transposes [2] Inverses [3] Determinants [] Transposes The transpose of a matrix A is denoted A T, or in Matlab, A. The transpose of a matrix exchanges the rows and
More informationThe matrix equation Ax = b can be written in the equivalent form
Last lecture (revision) Let A = (a ij ) be an n mmatrix and let a i = be column i of A ai a ni The matrix equation Ax = b can be written in the equivalent form x a + x 2 a 2 + + x m a m = b Claim The
More informationMath 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
More informationSolution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.
Solutions to Math 30 Takehome prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)
More informationfor any pair of vectors u and v and any pair of complex numbers c 1 and c 2.
Linear Operators in Dirac notation We define an operator Â as a map that associates with each vector u belonging to the (finitedimensional) linear vector space V a vector w ; this is represented by Â u
More information1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)
Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible
More information(u, Av) = (A T u,v), (6.4)
216 SECTION 6.1 CHAPTER 6 6.1 Hermitian Operators HERMITIAN, ORTHOGONAL, AND UNITARY OPERATORS In Chapter 4, we saw advantages in using bases consisting of eigenvectors of linear operators in a number
More informationCoordinates. Definition Let B = { v 1, v 2,..., v n } be a basis for a vector space V. Let v be a vector in V, and write
MATH10212 Linear Algebra Brief lecture notes 64 Coordinates Theorem 6.5 Let V be a vector space and let B be a basis for V. For every vector v in V, there is exactly one way to write v as a linear combination
More informationMath 480 Diagonalization and the Singular Value Decomposition. These notes cover diagonalization and the Singular Value Decomposition.
Math 480 Diagonalization and the Singular Value Decomposition These notes cover diagonalization and the Singular Value Decomposition. 1. Diagonalization. Recall that a diagonal matrix is a square matrix
More informationA Linear Algebra Primer James Baugh
Introduction to Vectors and Matrices Vectors and Vector Spaces Vectors are elements of a vector space which is a set of mathematical objects which can be added and multiplied by numbers (scalars) subject
More information1. Linear systems of equations. Chapters 78: Linear Algebra. Solution(s) of a linear system of equations. Row operations.
A linear system of equations of the form Sections 75 78 & 8 a x + a x + + a n x n = b a x + a x + + a n x n = b a m x + a m x + + a mn x n = b m can be written in matrix form as AX = B where a a a n x
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationMath 215 Exam #1 Solutions
Math 25 Exam # Solutions. (8 points) For each of the following statements, say whether it is true or false. Please write True or False and not just T or F (since these letters are easily mistaken for each
More informationMath 2040: Matrix Theory and Linear Algebra II Solutions to Assignment 4
. Complex Eigenvalues Math 00: Matrix Theory and Linear Algebra II Solutions to Assignment... Problem [ Restatement: ] Find the eigenvalues and a basis of the eigenspace in C of A =. Final Answer: The
More informationMath 2040: Matrix Theory and Linear Algebra II Solutions to Assignment 3
Math 24: Matrix Theory and Linear Algebra II Solutions to Assignment Section 2 The Characteristic Equation 22 Problem Restatement: Find the characteristic polynomial and the eigenvalues of A = Final Answer:
More information(a) If A is an n n matrix with nonzero determinant and AB = AC then B = C. (b) A square matrix with zero diagonal entries is never invertible.
1. or : (a) If A is an n n matrix with nonzero determinant and AB = AC then B = C. (b) A square matrix with zero diagonal entries is never invertible. (c) A linear transformation from R n to R n is onetoone
More information(2) Show that two symmetric matrices are similar if and only if they have the same characteristic polynomials.
() Which of the following statements are true and which are false? Justify your answer. (a) The product of two orthogonal n n matrices is orthogonal. Solution. True. Let A and B be two orthogonal matrices
More informationLinear Algebra Review (with a Small Dose of Optimization) Hristo Paskov CS246
Linear Algebra Review (with a Small Dose of Optimization) Hristo Paskov CS246 Outline Basic definitions Subspaces and Dimensionality Matrix functions: inverses and eigenvalue decompositions Convex optimization
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationMath 220 Sections 1, 9 and 11. Review Sheet v.2
Math 220 Sections 1, 9 and 11. Review Sheet v.2 Tyrone Crisp Fall 2006 1.1 Systems of Linear Equations Key terms and ideas  you need to know what they mean, and how they fit together: Linear equation
More informationMath 215 Exam #1 Practice Problem Solutions
Math 5 Exam # Practice Problem Solutions For each of the following statements, say whether it is true or false If the statement is true, prove it If false, give a counterexample (a) If A is a matrix such
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Coordinates and linear transformations (Leon 3.5, 4.1 4.3) Coordinates relative to a basis Change of basis, transition matrix Matrix
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationMath Final Review Dec 10, 2010
Math 301001 Final Review Dec 10, 2010 General Points: Date and time: Monday, December 13, 10:30pm 12:30pm Exam topics: Chapters 1 4, 5.1, 5.2, 6.1, 6.2, 6.4 There is just one fundamental way to prepare
More informationExample Linear Algebra Competency Test Solutions. N/A vectors and matrices must be of the same dimensions for addition to be defined.
Example Linear Algebra Competency Test Solutions The 40 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter
More informationA Crash Course in Linear Algebra
A Crash Course in Linear Algebra Jim Fakonas October, 202 Definitions The goal of this section is to provide a brief refresher in the basic terms and concepts of linear algebra, listed here roughly in
More informationReview: Vector space
Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space.
More informationMatrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: x n.
X. LINEAR ALGEBRA: THE BASICS OF MATRICES Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: y = a 1 + a 2 + a 3
More informationLinear Algebra and Matrices
LECTURE Linear Algebra and Matrices Before embarking on a study of systems of differential equations we will first review, very quickly, some fundamental objects and operations in linear algebra.. Matrices
More informationQuestions on Eigenvectors and Eigenvalues
Questions on Eigenvectors and Eigenvalues If you can answer these questions without any difficulty, the question set on this portion within the exam should not be a problem at all. Definitions Let A be
More information2.5 Spaces of a Matrix and Dimension
38 CHAPTER. MORE LINEAR ALGEBRA.5 Spaces of a Matrix and Dimension MATH 94 SPRING 98 PRELIM # 3.5. a) Let C[, ] denote the space of continuous function defined on the interval [, ] (i.e. f(x) is a member
More informationINTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL
SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics
More informationLinear Algebra Summary
Linear Algebra Summary 1. Linear Equations in Linear Algebra 1.1 Definitions and Terms 1.1.1 Systems of Linear Equations A linear equation in the variables x 1, x 2,..., x n is an equation that can be
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) May 12, 2015
Final Exam Advanced Linear Algebra Math 4377 / 638 (Spring 215) May 12, 215 4 points 1. Label the following statements are true or false. (1) If S is a linearly dependent set, then each vector in S is
More informationLinear Algebra PRACTICE EXAMINATION SOLUTIONS
Linear Algebra 2S2 PRACTICE EXAMINATION SOLUTIONS 1. Find a basis for the row space, the column space, and the nullspace of the following matrix A. Find rank A and nullity A. Verify that every vector in
More informationMath 1180, Hastings. Notes, part 9
Math 8, Hastings Notes, part 9 First I want to recall the following very important theorem, which only concerns square matrices. We will need to use parts of this frequently. Theorem Suppose that A is
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationExamination in TMA4110/TMA4115 Calculus 3, August 2013 Solution
Norwegian University of Science and Technology Department of Mathematical Sciences Page of Examination in TMA40/TMA45 Calculus 3, August 03 Solution 0 0 Problem Given the matrix A 8 4. 9 a) Write the solution
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationGRE math study group Linear algebra examples D Joyce, Fall 2011
GRE math study group Linear algebra examples D Joyce, Fall 20 Linear algebra is one of the topics covered by the GRE test in mathematics. Here are the questions relating to linear algebra on the sample
More informationPOL502: Linear Algebra
POL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Definition 1 A m n matrix A is a rectangular array of numbers with
More informationDifference Equations. next page. close. exit. Math 45 Linear Algebra. David Arnold.
Math 45 Linear Algebra David Arnold DavidArnold@Eureka.redwoods.cc.ca.us Abstract This activity investigates the form of d form solutions of first order difference equations of the form u k = Au k where
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationLECTURE NOTES FOR 416, INNER PRODUCTS AND SPECTRAL THEOREMS
LECTURE NOTES FOR 416, INNER PRODUCTS AND SPECTRAL THEOREMS CHARLES REZK Real inner product. Let V be a vector space over R. A (real) inner product is a function, : V V R such that x, y = y, x for all
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationMATH 33A LECTURE 2 FINAL EXAM #1 #2 #3 #4 #5 #6. #7 #8 #9 #10 #11 #12 Total. Student ID:
MATH A LECTURE FINAL EXAM Please note: Show your work. Except on true/false problems, correct answers not accompanied by sufcent explanations will receive little or no credit. Please call one of the proctors
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationTo provide background material in support of topics in Digital Image Processing that are based on matrices and/or vectors.
Review Matrices and Vectors Objective To provide background material in support of topics in Digital Image Processing that are based on matrices and/or vectors. Some Definitions An m n (read "m by n")
More informationMath Winter Final Exam
Math 02  Winter 203  Final Exam Problem Consider the matrix (i) Find the left inverse of A A = 2 2 (ii) Find the matrix of the projection onto the column space of A (iii) Find the matrix of the projection
More informationVector Spaces and Linear Transformations
Vector Spaces and Linear Transformations Beifang Chen Fall 6 Vector spaces A vector space is a nonempty set V whose objects are called vectors equipped with two operations called addition and scalar multiplication:
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25Sep02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationLinear Algebra Prerequisites  continued. Jana Kosecka
Linear Algebra Prerequisites  continued Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html kosecka@cs.gmu.edu Matrices meaning m points from ndimensional space n x m matrix transformation Covariance
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationSolution Set 8, Fall 11
Solution Set 8 186 Fall 11 1 What are the possible eigenvalues of a projection matrix? (Hint: if P 2 P and v is an eigenvector look at P 2 v and P v) Show that the values you give are all possible Solution
More informationMore Linear Algebra Study Problems
More Linear Algebra Study Problems The final exam will cover chapters 3 except chapter. About half of the exam will cover the material up to chapter 8 and half will cover the material in chapters 93.
More informationMATH10212 Linear Algebra B Homework 7
MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments
More informationApplication of Linear Algebra on Least Squares Approximation
Application of Linear Algebra on Least Squares Approximation Kelan Lu Doctoral Student Univ. of North Texas Dept. of Political Science lillylu01@gmail.com May 8, 2010 An Introduction of the Least Squares
More information