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Math 235 Midterm 1 Spring 1999

  1. a) Find the row reduced echelon augmented matrix of the system

    b) Find the general solution for the system.

  2. Determine if the statements are true or false. If it is true, give a reason. If it is false, provide a counter example:

    1. tex2html_wrap_inline109
    2. The columns of any tex2html_wrap_inline111 matrix are linearly dependent.
    3. If a set in tex2html_wrap_inline113 is linearly dependent, then the set contains more than n vectors.
    4. If the system tex2html_wrap_inline117 has a unique solution, then the columns of the matrix A are linearly independent.
    5. If the columns of a tex2html_wrap_inline121 matrix A are linearly independent, then the system tex2html_wrap_inline125 is consistent for every vector tex2html_wrap_inline127 in tex2html_wrap_inline129 .
    6. Let A and B be two tex2html_wrap_inline135 matrices and C=AB their product. If the third column tex2html_wrap_inline139 of B is a linear combination of the first two columns tex2html_wrap_inline143 , tex2html_wrap_inline145 , then the third column tex2html_wrap_inline147 of C is a linear combination of the first two columns tex2html_wrap_inline151 , tex2html_wrap_inline153 .
    7. If tex2html_wrap_inline155 is a linearly dependent set in tex2html_wrap_inline157 , then the vector tex2html_wrap_inline159 is a linear combination of tex2html_wrap_inline161 and tex2html_wrap_inline163 .

  3.   Set up a system of linear equations for finding the electrical currents tex2html_wrap_inline165 in the following circuit using i) the junction rule: the sum of currents entering a junction is equal to the sum of currents leaving the junction. ii) Ohm's rule: The drop in the voltage tex2html_wrap_inline167 across a resistance R is related to the (directed) current I by the equation tex2html_wrap_inline173 . iii) Kirchhof's circuit rule: the sum of the voltage drops due to resistances around any closed loop in the circuit equals the sum of the voltages induced by sources along the loop.

    All resistances below are equal to tex2html_wrap_inline175 . Note: Do not solve the system.


  4. For each of the following maps, determine if it is a linear transformation. If it is, find its standard matrix. If it is not, explain which property of linear transformations it violates (and why it violates it).

    a) T is the map from tex2html_wrap_inline179 to tex2html_wrap_inline179 defined by tex2html_wrap_inline183 .

    b) T is the map from tex2html_wrap_inline179 to tex2html_wrap_inline129 defined by tex2html_wrap_inline191 .

    c) T is the map from tex2html_wrap_inline179 to tex2html_wrap_inline179 which is the composition of the rotation of the plane 90 degrees counterclockwise, followed by reflection with respect to the tex2html_wrap_inline201 -axis.

  5. For each of the following sets of vectors in tex2html_wrap_inline113 answer both questions:
    i) Does the set span the whole of tex2html_wrap_inline113 ? and ii) Is it linearly independent? Justify your answer!!! (No credit will be given for an answer without a justification).

    1. tex2html_wrap_inline207 , tex2html_wrap_inline209 , tex2html_wrap_inline211
    2. tex2html_wrap_inline213 , tex2html_wrap_inline215
    3. tex2html_wrap_inline217 , tex2html_wrap_inline219 , tex2html_wrap_inline221
  6. Let tex2html_wrap_inline223 , tex2html_wrap_inline225 , and C=AB their product.

    a) Compute the (3,2) entry of C.

    b) Let tex2html_wrap_inline153 be the second column of C. Solve (with as little computations as possible) the system of linear equations tex2html_wrap_inline237 .

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Eyal Markman
Thu Sep 28 15:33:09 EDT 2000