MAT 207 LINEAR ALGEBRA I 2009207 Dokuz Eylül University, Faculty of Science, Department of Mathematics Instructor: Engin Mermut web: http://kisideuedutr/enginmermut/ HOMEWORK VECTORS IN THE n-dimensional SPACE R n, AND SYSTEMS OF LINEAR EQUATIONS Textbook: Linear Algebra: A Geometric Approach, Theodore Shifrin and Malcolm R Adams, 2nd edition, W H Freeman and Company, 20 SOLVE THE BELOW EXERCISES FROM YOUR TEXTBOOK Of course, solve as many exercises as you need to be sure that you have learned the concepts, the precise definitions and you can write proofs of the asked results Be sure that you can do the computational exercises without error You must understand the algorithms given in the proofs of the theorems, this will help you understand the theorems and get a feeling why the results of these theorems are true Be prepared to solve some of the following exercises in the next weeks: Section Vectors (pages 5 8): Exercises 6, 8 25, 28, 29 Section 2 Dot Product (pages 25 28): Exercises 2, 5, 7, 9,, 6 9, 2, 22, 24, 25 Section Hyperplanes in R n (pages 4 6): Exercises 5 9, Section 4 Systems of Linear Equations and Gaussian Elimination (pages 49 5): Exercises 7 You must do all of the exercises of this and the next section Obtaining an echelon form of any matrix by Gaussian elimination, and then obtaining the unique reduced echelon form of a matrix is one of the fundamental topics of your course that you must understand and do well This is also the subject for your quiz; your quiz problem will be like the exercises of this section and the next section Section 5 The Theory of Linear Systems (pages 6 64): Exercises 5 DEFINITIONS Write the precise definition of the following terms Let n, m be positive integers When do we say that two vectors x and y in R n are parallel vectors? Nonparallel vectors? 2 What is the parametric form and Cartesian form to define the following geometric objects: (a) Lines in R 2 (b) Planes in R (c) Lines in R What is a normal vector to a line in R 2? What is a normal vector to a plane in R? 4 What is a line in R n? A plane in R n? A hyperplane in R n? 5 What is a normal vector to a hyperplane in R n? 6 When do we say that three or more points in R n are collinear? 7 Let k Z + and let v, v 2,, v k be k vectors in the n-dimensional space R n (a) What is a linear combination of these k vectors v, v 2,, v k?
(b) What is the span of these k vectors v, v 2,, v k? (c) What does Span(v, v 2,, v k ) denote? 8 What is the dot product of two vectors in R n? When do we say that two vectors in R n are orthogonal? What is the definition of the angle between two nonzero vectors in R n? 9 What is the projection of a vector x in R n onto a nonzero vector y in R n? proj y x =? 0 What is the distance from the origin to a hyperplane in R n? What does Cauchy-Schwarz inequality say for two vectors in R n? When does equality occurs in Cauchy-Schwarz inequality? 2 What is a system of m linear equations in n variables? What is the definition of the solution set of such a system? What is the augmented matrix associated to such a system? What are the elementary operations on a system of m linear equations in n variables? 4 What are the elementary row operations on matrices? 5 What is the leading entry of a nonzero row of a matrix? 6 What is the precise definition for an m n matrix A to be in reduced echelon form? Is the following definition correct: An m n matrix A = [a ij ] m,n i,j= is in reduced echelon form if either it is the zero matrix (all entries are zero), or there exists a positive integer r that is m, and r positive integers k, k 2,, k r such that k i n for i =, 2,, m, and for all integers i, j with i m and j n, the following hold: (a) k < k 2 < < k r (b) a ij = 0 if j < k i ; {, if i = j; (c) a i,kj = δ ij = 0, if i j That is, a i,ki = and a i,kj = 0 if j i (d) a ij = 0 if i > r; 7 What is the precise definition for an m n matrix A to be in echelon form? Give the definition like the form suggested in the previous question a a 2 a n b 8 Let A = [a ij ] m,n i,j= = a 2 a 22 a 2n be an m n matrix in echelon form Let b = a m a m2 a mn b m x be in R m Consider the system of equations Ax = b for x = in Rn : a x + a 2 + + a n = b a 2 x + a 22 + + a 2n = a m x + a m2 + + a mn = b m (a) What are pivots? (b) What are pivot columns? (c) What are pivot variables? (d) What are free variables? 2
9 What is Gaussian elimination? 20 What is an echelon form of a matrix A? What is the reduced echelon form of a matrix A? 2 What is the rank of an m n matrix? 22 What is a consistent system of linear equations? Inconsistent? a a 2 a n 2 Let A = [a ij ] m,n i,j= = a 2 a 22 a 2n be an m n matrix Let x = a m a m2 a mn constraint equations that describe the vectors b = b b m x What are the in Rm for which the system Ax = b is consistent (that is b can be expressed as a linear combination of the columns of A)? 24 What is a homogeneous system of linear equations? 25 What is an inhomogeneous system of linear equations? What is the associated homogeneous system to an inhomogeneous system? What is a particular solution of the inhomogeneous system? 26 What is a nonsingular matrix? What is a singular matrix? TRUE OR FALSE Prove your answer You must prove it if it is true If it is false, you must give a counterexample or show why it is false; again a proof is required of course Let n, m Z + The below questions are for the n-dimensional space R n and matrices whose entries are real numbers A line l in R n is a set of the form where x 0 and v are in R n 2 A plane P in R n is a set of the form where x 0, u and v are in R n The following set P is a plane in R : l = {x R n x = x 0 + tv for some t R} P = {x R n x = x 0 + su + tv for some s, t R} P = {x R x = (7, 4, ) + s(, 6, 9) + t(5, 0, 5) for some s, t R} 4 For all vectors x R n, x x = x 2 5 For all vectors x R n, x x 0 6 For all vectors x R n, x x = 0 if and only if x is the zero vector 7 For all vectors x, y R n, x y = y x 8 For all vectors x, y R n, and for all c R, (cx) y = c(x y) 9 For all vectors x, y, z R n, x (y + z) = x y + x z 0 For all vectors x, y in R n, x + y 2 = x 2 2x y + y 2
The projection of a vector x in R n onto a nonzero vector y in R n is the unique vector z R n such that z = cy for some real number c and x z is orthogonal to y 2 The projection of a vector x in R n onto a nonzero vector y in R n is the vector ( ) x y proj y x = y y For all vectors x, y in R n, x + y 2 = x 2 + y 2 if and only if x and y are orthogonal vectors 4 For all vectors x, y in R n, where y 0, the vector x ( ) x y y 2 y is orthogonal to y 5 For all vectors x, y in R n, if y is a unit vector, then x (x y)y is orthogonal to y, and x = (x y)y + [x (x y)y] 6 The angle between two nonzero vectors x and y in R n is ( ) x y θ = arccos x y 7 The distance from the origin to the hyperplane in R n with normal vector a is a x 0 a for any point x 0 on the plane 8 The point closest to the origin on the hyperplane in R n with normal vector a is the point a x 0 a 2 a where x 0 is any point on the plane 9 For each m n matrix A, by performing elementary row operations, we obtain a matrix in echelon form 20 For each m n matrix A, by performing elementary row operations, we obtain a matrix in reduced echelon form 2 Every m n matrix A has a unique echelon form 22 Every m n matrix A has a unique reduced echelon form 2 Solving a system of m linear equations in n variables, where in each equation at least one variable has a nonzero coefficient, is equivalent to finding the intersection of m hyperplanes in R n 24 The rank of a matrix A is the number of nonzero rows in any echelon form of A 25 The rank of a matrix A is the number of pivots in any echelon form of A 26 A system of equations Ax = b has a unique solution if and only if the associated homogeneous system Ax = 0 has only the trivial solution 0 27 For a square matrix A, A is nonsingular if and only if Ax = 0 has only the trivial solution 0 28 An n n square matrix A is nonsingular if and only if for every b R n, the equation Ax = b has a solution 29 A n n square matrix A is nonsingular if and only if for every b R n, the equation Ax = b has a unique solution 0 For an m n matrix A, the system of equations Ax = b has a unique solution if and only if the rank r of A equals n 4
If A is an m n matrix with rank m and v, v 2,, v k R n are vectors such that then Span(Av, Av 2,, Av k ) = R m Span(v, v 2,, v k ) = R n, 2 A system of linear equations is either inconsistent or has a unique solution If a system of linear equations is consistent, then it has infinitely many solutions 4 If the number of variables in a system of linear equations is less than the number of equations, then the system is either inconsistent or has infinitely many solutions 5 If the number of variables in a homogeneous system of linear equations is less than the number of equations, then the homogeneous system has a nontrivial solution, that is a solution different from the trivial solution 0 6 If in a system of linear equations, the number of equations is equal to the number of variables, then it has a unique solution if and only if the associated homogeneous system has no solutions other than the trivial solution 0 REVIEW QUESTIONS Be sure that you know the answers to the following questions; review your lecture notes and see your textbook to learn the related concepts: What are the "k-dimensional" "straight objects" in the n-dimensional space R n (where k n are positive integers)? What do you need to have really k dimensions? What is your intuitive idea to have k dimensions? Do you know the precise definition of a k-dimensional subspace of R n? What do you think "subspace" should mean? 2 How does Cauchy-Schwarz inequality enable us to define the angle between two nonzero vectors in R n? What is a hyperplane in R n? How many "dimensions" does a hyperplane in R n have? What is the Cartesian equation for a hyperplane in R n? What is the parametric description of a hyperplane in R n? 4 Does the solution set of a system of m linear equations in n variables change under elementary operations on equations? 5 What is the shape of a matrix in echelon form? In reduced echelon form? What is the difference between a matrix in echelon form and a matrix in reduced echelon form? 6 Does every matrix have an echelon form? Is it unique? If so, prove it, or give a counterexample 7 Does a matrix have a reduced echelon form? Is it unique? If so, prove it This is an important problem to work on 8 If B is an echelon form of a matrix A, by doing which operations on B and in which order, do you obtain the reduced echelon form of A? 9 Does all echelon forms of an m n matrix A have the same number of nonzero rows? a a 2 a n b 0 Let A = [a ij ] m,n i,j= = a 2 a 22 a 2n be an m n matrix and let b = be in Rm a m a m2 a mn b m x How do we give a parametric description of all solutions x = in Rn to the system Ax = b by a a 2 a n b a 2 a 22 a 2n finding the reduced echelon form of the augmented matrix [A b] = a m a m2 a mn b m by Gaussian elimination? 5
a a 2 a n Let A = [a ij ] m,n i,j= = a 2 a 22 a 2n be an m n matrix whose all rows are nonzero (that a m a m2 a mn is, every row of A has at least one nonzero entry) Let b = How do we interpret the solution set of the system Ax = b for x = b b m be in Rm x in Rn as the intersection of m hyperplanes in R n whose normal vectors are obtained from the row vectors of the matrix A? Remember that we denote the row vectors of the m n matrix A = [a ij ] m,n i,j= by A i = [ a i a i2 a in ] for i =, 2,, m a a 2 a n x 2 Let A = [a ij ] m,n i,j= = a 2 a 22 a 2n be an m n matrix Let x = be in Rn How a m a m2 a mn do we interpret Ax as a linear combination of the n columns a, a 2,, a n of the m n matrix A with coefficients x,,,? Remember that we denote the column vectors of the m n matrix A = [a ij ] m,n i,j= by Let b = b b m a j = a i a i2 a im for j =, 2,, n be in Rm Explain why it is true that b is a linear combination of the n columns a, a 2,, a n of the matrix A if and only if the system Ax = b has a solution for x = How do we find the constraint equations for b = b x in Rn in Rm to be a linear combination of the n columns a, a 2,, a n of the matrix A? b m One of the main themes you must understand well is the passage between a parametric description of a set of vectors in R n and its description as the solution set of a system of linear Cartesian equations Explain how do you pass from one form to the other using the methods of solving systems of linear equations More precisely: (a) If you are given m linear Cartesian equations in n variables, how do you give the parametric description of the solution set in R n? How many free parameters are used and so what seems to be the dimension of the solution set? That is, if the solution set is a k-dimesional straight object in R n, how dou you determine k? 6
(b) Let k Z + and let v, v 2,, v k be k vectors in the n-dimensional space R n Let x 0 be also a vector in R n For the following set given in parametric form, how do you give its description by a system of m linear Cartesian equations in n variables? {x 0 + u u Span(v, v 2,, v k )} = {x 0 + t v + t 2 v 2 + + t k v k t, t 2,, t k R} For this set to a k-dimesional straight object in R n, what do you need to assume for the vectors v, v 2,, v k? 4 Using an echelon form of the augmented matrix [A b] of a system of linear equations Ax = b: (a) How do you determine if it is consistent, that is, if it has a solution? (b) How do you determine if it is inconsistent, that is, if it has no solution? (c) How do you determine if it it has a unique solution? Is it related with the rank r of the matrix A? (d) How do you determine if the matrix A is nonsingular? PROBLEMS Let f : R n R be a function Prove that the following are equivalent for the function f : R n R: (a) For all x, y in R n and c R, f(x + y) = f(x) + f(y) and f(cx) = cf(x) (b) There exist a, a 2,, a n R such that for all x = x in Rn, f(x) = a x + a 2 + + a n 2 Let A be an m n matrix and b R m Prove by following the below argument that if a sytem Ax = b is consistent and does not have a unique solution, then it has infinitely many solutions If u and v are two distinct solutions in R n of the system Ax = b, then prove that for every t R, tu + ( t)v is also a solution of the sytem Ax = b Show that these give infinitely many distinct solutions by proving that for all s, t R, if s t, then tu + ( t)v su + ( s)v In a homogenus sytem of linear equations, if the number of variables is greater than the number of equations, then show that the homogeneous sytem has a nontrivial solution, that is, a solution different from the trivial solution 0 4 Let A be an n n square matrix and b R n So in the sytem Ax = b, the number of variables is equal to the number of equations Prove that the property of having a unique solution depends only on the matrix A; it does not depend on the vector b 5 Let A be an m n matrix and b R m Consider the system Ax = b consisting of m linear equations in n variables x,,, If the number of variables is greater than the number of equations, that is, if n > m, then prove that the system is either inconsistent or has infinitely many solution 6 Given n+ distinct real numbers c, c 2,, c n+ and any real numbers b,,, b n+, does there exist a polynomial f(x) = a 0 + a x + a 2 + + a n of degree at most n with real number coefficients a 0, a,, a n such that f(c i ) = b i for i =, 2,, n, n +? Hint: Do we have a system of linear equations for the n + unknowns a 0, a,, a n? How many equations? When does it have a unique solution? 7
SAMPLE QUIZ PROBLEMS Find all solutions in real numbers of the following systems of linear equations: 2x 7x +5x 4 +2x 5 = 2 x 2 4x +x 4 +x 5 = 2 2x 4x +2x 4 +x 5 = x 5 7x +6x 4 +2x 5 = 7 2 Consider the following four vectors in R 4 : v = 2 0, v 2 = (a) Is b = (b) Is b = 0 7 4 2 in Span(v, v 2, v, v 4 )? in Span(v, v 2, v, v 4 )? 6 4 0 2 (c) Find the constraint equations that a vector b =, v = b b b 4 2, v 4 = combination of v, v 2, v, v 4 (that is, to be in Span(v, v 2, v, v 4 )) b (d) If b = b is in Span(v, v 2, v, v 4 ), find all x,, x, x 4 such that b 4 b = x v + v 2 + x v + x 4 v 4 0 in R4 must satisfy in order to be a linear Find a normal vector to the hyperplane in R 4 spanned by the three vectors v = (,,, ), v 2 = (, 2,, 2), v = (,, 2, 4), 4 Consider the following four hyperplanes in R 5 with the given normal vectors and a point on them: Hyperlane a normal vector to the hyperplane a point on the hyperplane P a = (, 2,, 2, ) b = (0,, 0, 0, 0) P 2 a 2 = (, 2,, 2, ) = (0, 0, 0, 0, 2) P a = (2, 4,, 2, 0) b = (0, 0,, 0, 0) P 4 a 4 = (, 6, 2, 0, ) b 4 = (, 0, 0, 0, 0) Find the intersection of these four hyperplanes in R 5 ; give a parametric description of the points in the intersection 8