Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath 45—Linear AlgebraPretest–Exam #4Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: December 1, 2000 Version 1.002Essay QuestionsDirections: Place the solution to each of the following exercises onyour own paper. You must follow directions explicitly and show allwork to receive full credit.Exercise 1. Find(a) the projection of the vector (2, 1) onto the vector (1, 1), and(b) find the projection matrix P that will project all vectors in theplane onto the line through the origin that points in the direction(1, 1),Exercise 2. In 3-space,(a) find the equation of the line passing through the origin in thedirection of (1, 1, 1), and(b) find the equation of the plane passing through the origin that isperpendicular to the line in part (a).3Exercise 3. What is the shortest distance between the line y =2x+3and the point (5, −2)?Exercise 4. Find the point on the line y =1− 2x that is closest tothe point (3, 1).4Exercise 5. Find the distance between the point (1, 2, −1) and theplane x +2y +3z =0.5Exercise 6. Consider the plane in R3spanned by the vectors (1, 1, 0)Tand (0, 1, 1)T.(a) Find the projection matrix P that will project all vectors in R3onto the plane defined above.(b) Find the projection of the vector b =(1, 2, −3)Tonto the planedefined above.Exercise 7. Consider the matrix1 −10000011100000Find bases for the four fundamental subspaces. Place your results ona Strang diagram.6Exercise 8. Find the eigenvalues and eigenvectors of each matrix.(a)7 −105 −8(b)5 −34020−88−77Exercise 9. Consider the following table of data points.xy0 51121(a) Plot the data on graph paper.(b) Find the equation of the line of best fit (hand calculations only)without the aid of technology.(c) Draw the line of best fit on your graph paper,(d) Find the sum of the squares of the errors made.Exercise 10. Prove that similar matrices have the same eigenvalues.Exercise 11. Let R be a 2 × 2 matrix that reflects vectors across agiven line in the plane. Use the geometry of the situation to find theeigenvalues of the matrix.8Exercise 12. Consider the matrix1 −22 −4(a) Find the 4 fundamental subspaces.(b) Draw N(A)andC(AT) in the plane.(c) Draw C(A)andN (AT) in the plane.9Exercise 13. If we remember our theory of linear transformations,we know that any transformation P : R2→ R2is completely deter-mined by its action on the “standard basis” vectors. Let P be thematrix that projects vectors in the plane onto the line spanned by thevector (−1, 2)T.(a) The formula for projecting vector b onto a isp =b · aa · aaForm the matrix P , where the first column of P is the projectionof e1=(1, 0)Tonto a =(−1, 2) and the second column of P isthe projection of e2=(0, 1)Tonto a =(−1, 2)T.(b) Use the formulaP =aaTaTato compute the matrix P that projects vectors in the plane ontothe line spanned by a =(−1, 2)T. Compare this with the resultfound in part (a).10Exercise 14. Find a matrix A that hasB =1100,0111as a basis for its column space andB =111,100as a basis for its row space (C(AT)).Exercise 15. Find a matrix whose row space is spanned by (1, 1, 2)Tand whose null space is spanned by(a) (1, 2, −1)T.(b) (1, 2, −1)T.11Exercise 16. If A is a 7 × 9 matrix with nullity 2, what are the di-mensions of the 4 fundamental subspaces? Include a Strang diagram.Here are a few fun questions from Professor Strang’s MIT quizzes.Exercise 17. SupposeA =1001107 −12101450122100011.(a) What is the rank of A?(b) Find a basis for the nullspace of A.(c) Find the complete solution toAx =10158512Exercise 18. Suppose that row operations (elimination) reduce thematrices A and B to the same row echelon formR =120700150000.(a) Which of the four subspaces are sure to be the same for A andB?(C(A)=C(B)? N (A)=N (B)? C(AT)=C(BT)? N (AT)=N(BT)?)(b) Each time the subspaces in part (a) are the same for A and B,find a basis for that subspace.(c) True or False (A is any matrix and x, y are two vectors): IfAx and Ay are linearly independent then x and y are linearlyindependent.13Exercise 19. Suppose A is an m by n of rank r.(a) If Ax = b has a solution for ever right side b, what is the columnspace of A?(b) In part (a), what are all equations or inequalities that must holdbetween the numbers m, n,andr.(c) Give a specific example of a 3 by 2 matrix A of rank 1 with firstrow [2 5]. Describe the column space C(A) and the nullspaceN(A) completely.(d) Suppose the right side b is the same as the first column in yourexample (part c). Find the complete solution to Ax = b.14Exercise 20. Let A by an n × n matrix.(a) If the row space of A is Rnthen the column space of A is ?(b) If the nullspace of A is Rnthen the column space of A is ?(c) If the left nullspace of A is Rnthen the column space of A is ?(d) Give an example of a square matrix A such that the column spaceis orthogonal to the row space.(e) If the column space of an n × n matrix A is orthogonal to the rowspace there is an inequality relating the rank r to n. What is thestrongest possible inequality? (Hint: r ≤ n is a true inequality,but is not the strongest and hence will be considered an incorrectanswer. Only the right answer will be given credit.)(f) If the column space is orthogonal to the row space, then det(A)=?Solutions to Exercises 15Solutions to ExercisesExercise 1(a) Solutions to Exercises 16Exercise 1(b) Solutions to Exercises 17Exercise 2(a) Solutions to Exercises 18Exercise 2(b) Solutions to Exercises 19Exercise 3. Exercise 3Solutions to Exercises 20Exercise 4. Exercise 4Solutions to Exercises 21Exercise 5. Exercise 5Solutions to Exercises 22Exercise 6(a) Solutions to Exercises 23Exercise 6(b) Solutions to Exercises 24Exercise 7. Exercise 7Solutions to Exercises 25Exercise 8(a) Solutions to Exercises 26Exercise 8(b) Solutions to Exercises 27Exercise 9(a) Solutions to Exercises 28Exercise 9(b) Solutions to Exercises 29Exercise 9(c) Solutions to Exercises 30Exercise 9(d) Solutions to Exercises 31Exercise 10. Exercise 10Solutions to Exercises 32Exercise 11. Exercise 11Solutions to Exercises 33Exercise 12(a) Solutions to Exercises 34Exercise 12(b) Solutions to Exercises