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Math 235 Midterm 1 solution Fall 2000

  1. (20 points) a) Find the row reduced echelon augmented matrix of the system

    Answer: The row reduction takes four steps:


    b) Find the general solution for the system.

    Answer: tex2html_wrap_inline214 and tex2html_wrap_inline216 are free variables.


  2. a) (12 points) For which values of h will the vector tex2html_wrap_inline222 be in the plane passing through tex2html_wrap_inline224 , tex2html_wrap_inline226 , and the origin (zero vector).

    Answer: The question is equivalent to:

    ``For which value of h is w in the plane spanned by tex2html_wrap_inline232 and tex2html_wrap_inline234 ?''

    The latter question is equivalent to:

    ``For which h is the system, corresponding to the following augmented matrix, consistent?''


    If h=3, the system is consistent, and w is a linear combination of tex2html_wrap_inline232 and tex2html_wrap_inline234 (indeed, tex2html_wrap_inline248 ).

    If tex2html_wrap_inline250 , the system is inconsistent and w does not belong to tex2html_wrap_inline254 .

    b) (8 points) Find two vectors tex2html_wrap_inline232 , tex2html_wrap_inline234 which span the plane P given by the equation


    (i.e., such that tex2html_wrap_inline262 ).

    Answer: Once we write the solution of the system (of one equation) in parametric form, we exhibit the solution set (i.e., the plane P) as a ``span''. The variables tex2html_wrap_inline266 and tex2html_wrap_inline268 are free. We get


    The two vectors on the right span the plane P.

  3. a) Is the set tex2html_wrap_inline274 , tex2html_wrap_inline276

    i) (5 points) linearly independent?

    Answer: Yes, the two vectors are linearly independent. A row echelon matrix is tex2html_wrap_inline278 . It has a pivot in every column.

    ii) (5 points) Does it span tex2html_wrap_inline280 ? Justify you answers!

    Answer: No, the two vectors do not span tex2html_wrap_inline280 . We do not have a pivot in every row. (There are less vectors than entries in each vector).

    b) For which values of h will the vectors


    i) (5 points) be linearly independent?

    Answer: Never. Four vectors in tex2html_wrap_inline280 are always linearly dependent.

    ii) (5 points) span tex2html_wrap_inline280 ?

    Answer: Row reduce.


    If tex2html_wrap_inline292 , then we have a pivot in every row (and the four vectors span tex2html_wrap_inline280 ).

  4.   (10 points) Set up a system of linear equations for finding the electrical currents tex2html_wrap_inline296 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_inline298 across a resistance R is related to the (directed) current I by the equation tex2html_wrap_inline304 . 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.

    Note: Do not solve the system.

    Answer using branch currents: Using the junction rule we get:





    Using the Voltage loop law, we get:




    Answer using loop currents: (The loop currents, denoted tex2html_wrap_inline320 , tex2html_wrap_inline322 , tex2html_wrap_inline324 , correspond to the counterclockwise direction in each loop).




    A complete answer requires expressing the branch currents in terms of the loop currents: (no credit was deducted if you skipped this part).

    tex2html_wrap_inline332 , tex2html_wrap_inline334 , tex2html_wrap_inline336 , tex2html_wrap_inline338 , tex2html_wrap_inline340 , tex2html_wrap_inline342 .

  5. (12 points) Determine if the statement is true or false. If it is true, give a reason. If it is false, provide a counter example:

    1. Let tex2html_wrap_inline232 , tex2html_wrap_inline234 , tex2html_wrap_inline348 be three vectors in tex2html_wrap_inline350 . Then the set tex2html_wrap_inline352 is linearly dependent (regardless of the choice of tex2html_wrap_inline232 , tex2html_wrap_inline234 , tex2html_wrap_inline348 ).

      Answer: TRUE. Let tex2html_wrap_inline360 , tex2html_wrap_inline362 , and tex2html_wrap_inline364 . The three vectors tex2html_wrap_inline366 satisfy the linear relation


    2. The columns of any tex2html_wrap_inline368 matrix are linearly dependent.

      Answer: FALSE. Counter Example: tex2html_wrap_inline370

    3. If the columns of a tex2html_wrap_inline372 matrix span tex2html_wrap_inline280 , then they are linearly independent.

    Answer: TRUE. Since the columns span tex2html_wrap_inline280 , we have a pivot in every row. Since the matrix is a square matrix, it must also have a pivot in every column. Consequently, the columns are linearly independent.

  6. 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) (6 points) T is the map from tex2html_wrap_inline280 to tex2html_wrap_inline382 defined by tex2html_wrap_inline384 .

    Answer: A linear transformation with standard matrix tex2html_wrap_inline386

    b) (6 points) T is the map from tex2html_wrap_inline382 to tex2html_wrap_inline382 defined by tex2html_wrap_inline394 .

    Answer: Not a linear transformation. If it were linear, it would satisfy:

    (1) tex2html_wrap_inline396 , for any two vectors in tex2html_wrap_inline382 , and

    (2) tex2html_wrap_inline400 , for any scalar c and vecotr tex2html_wrap_inline404 .

    It fails to satisfy both (1) and (2). Take for example tex2html_wrap_inline406 and tex2html_wrap_inline408 . Then T((0,0)+(1,1))=T(1,1)=(3,4), while T(0,0)+T(1,1)=(1,3)+(3,4)=(4,7). So,


    and (1) is not satisfied. You can check that (2) fails if you take tex2html_wrap_inline406 and c=2.

    c) (6 points) T is the map from tex2html_wrap_inline382 to tex2html_wrap_inline382 which is the composition of reflection with respect to the tex2html_wrap_inline424 -axis followed by the rotation of the plane 90 degrees counterclockwise.

    Answer: A linear transformation. The reflection has the standard matrix tex2html_wrap_inline428 The rotation has the standard matrix tex2html_wrap_inline430

    Hence, tex2html_wrap_inline432 , while


    The standard matrix is tex2html_wrap_inline436

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Eyal Markman
Tue Oct 17 14:57:28 EDT 2000