linear algebra homework

| November 16, 2015

Math 2240                                                                         Exam1                                  Fall 2015

Name ____________________________________

Instructions: Please show all work in the spaces below, including showing and recording all steps for row reduction.


  1. Fill-in the blank spaces or circle the appropriate answer.
    1. Matrix is in row echelon form/reduced row echelon form/neither.                                                                                                                                                         [2 points]
      1. Circle all pivot positions in the matrix.    [2 points]


  1. List all basic variables __________ and free variables _________. [4 points]


  • Is the equation consistentwhen ? Why?    [4 points]






  1. The set , where and  is linearly _______________


because _________________________________________.                           [4 points]


  1. Write TRUE or FALSE. Justify. [4 points each]
    1. If equation has more than one solution, then equation  also has more than one solution.




  1. Suppose that a coefficient matrix for a system has four pivot columns, then the system is consistent.


  1. The homogeneous equation has the trivial solution if and only if the equation has at least one free variable.




  1. The columns of are linearly independent if the equation  has the trivial solution.




  1. If is a  matrix, then the range of the transformation  is .




  1. Given the system .
    1. Write the corresponding augmented matrix for the system, the row-reduce the matrix to row-echelon form. [4 points]






  1. Findall the values of and  such that the solution set of the system is
    1. Empty.                                                                                               [4 points]


  1. Contains a unique solution. [4 points]


  • Contains infinitely many solutions. [4 points]



  1. Given matrix and vector , write equation
    1. As a vector equation. [4 points]




  1. As a system of linear equations. [4 points]





  1. Consider the system associated to the augmented matrix.Write the general solution of the associated system in parametric vector form. [10 points]










  1. Consider the matrix .
    1. Row-reduce to reduced echelon form, recording all elementary row operations used.

[8 points]









  1. Are the columns of linearly independent? Why or why not?                                [4 points]





  1. Do the columns of span                                                                  [4 points]







  1. Let be a linear transformation such that ,  , , and . Find the image of  by the transformation .                           [8 points]















  1. The transformation is defined by Show that is not linear.                                                                                               [8 points]





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