Solutions to ExercisesCollege of the RedwoodsMathematics DepartmentMath 45—Linear AlgebraPretest–Exam #4Linear AlgebraDavid ArnoldCopyrightc 2000 [email protected] Revision Date: December 4, 2000 Version 1.002Essay QuestionsDirections: Place the solution to each of the following exercises on your own paper. You must followdirections explicitly and show all work to receive full credit.Exercise 1. Find(a) the projection of the vector (2, 1) onto the vector (1, 1), and(b) the projection matrix P that will project all vectors in the plane onto the line through the origin thatpoints in the direction (1, 1),Exercise 2. In 3-space,(a) find the equation of the line passing through the origin in the direction of (1, 1, 1), and(b) find the equation of the plane passing through the origin that is perpendicular to the line in part (a).Exercise 3. What is the shortest distance between the line y =2x + 3 and the point (5, −2)?Exercise 4. Find the point on the line y =1−2x that is closest to the point (3, 1).Exercise 5. Find the distance between the point (1, 2, −1) and the plane x +2y +3z =0.Exercise 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 plane defined above.Exercise 7. Consider the matrix1 −10000011100000Find bases for the four fundamental subspaces. Place your results on a Strang diagram.Exercise 8. Find the eigenvalues and eigenvectors of each matrix.(a)7 −105 −8(b)5 −34020−88−7Exercise 9. Consider the following table of data points.xy0 51 121(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.3(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 a given line in the plane. Use the geometryof the situation to find the eigenvalues of the matrix.Exercise 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.Exercise 13. If we remember our theory of linear transformations, we know that any transformation P :R2→ R2is completely determined by its action on the “standard basis” vectors. Let P be the matrix thatprojects vectors in the plane onto the line spanned by the vector (−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 projection of e1=(1, 0)Tonto a =(−1, 2) andthe second column of P is the 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 onto the line spanned by a =(−1, 2)T.Compare this with the result found in part (a).Exercise 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, 1, 3)T.Exercise 16. If A is a 7 ×9 matrix with nullity 2, what are the dimensions of the 4 fundamental subspaces?Include a Strang diagram.4Here 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Exercise 18. Suppose that row operations (elimination) reduce the matrices A and B to the same rowechelon formR =120700150000.(a) Which of the four subspaces are sure to be the same for A and B?(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): If Ax and Ay are linearly independent thenx and y are linearly independent.Exercise 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 column space of A?(b) In part (a), what are all equations or inequalities that must hold between the numbers m, n,andr.(c) Give a specific example of a 3 by 2 matrix A of rank 1 with first row [2 5]. Describe the column spaceC(A) and the nullspace N(A) completely.(d) Suppose the right side b is the same as the first column in your example (part c). Find the completesolution to Ax = b.Exercise 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 space is orthogonal to the row space.(e) If the column space of an n × n matrix A is orthogonal to the row space there is an inequality relatingthe rank r to n. What is the strongest possible inequality? (Hint: r ≤ n is a true inequality, but isnot the strongest and hence will be considered an incorrect answer. Only the right answer will be givencredit.)(f) If the column space is orthogonal to the row space, then det(A)=?Solutions to Exercises 5Solutions to ExercisesExercise 1(a) We use the formula for the projection of b =(2, 1)Tonto a =(1, 1)T.pabp =a · ba · aa=3211=3/23/2Exercise 1(b) The projection matrix is calculated withP =aaTaTa=121111=1/21/21/21/2Exercise 2(a) The vector−−→P0P must be a scalar multiple of a =(1, 1, 1)T.−−→P0P = λaxyz= λ111P0(0,0,0)P (x,y,z)aHence, the equation of the line, in parametric form isx = λy = λz = λSolutions to Exercises 6Exercise 2(b) Select a point P (x, y, z) in the plane. The vector a =(1, 1, 1)Tis orthogonal to the vector−−→P0P .aP0(0,0,0)P (x,y,z)Thus,−−→P0P · a =0xyx·111=0Therefore, the equation of the plane isx + y + z =0Exercise 3.apnbA vector in the direction of the line is a =(1, 2)T. A vector orthogonal to the line is u =(2, −1)T.Theshortest distance between the