Math 235 Solution for Midterm 1 Spring 1999

1. (15 points) a) The row reduced echelon augmented matrix of the system

is obtained as follows:

Note: The row reduction takes only 3 elementary row operations. If you needed more than three then you need to review the row reduction algorithm!

b) Two variables, , are free variables. The general solution of the system is

2. (35 points) Determine if the statements are true or false. If it is true, give a reason. If it is false, provide a counter example:

1. The statement is True. The two planes in are equal!
Reason: Denote by the two vectors on the left hand side, and by the two vectors on the right hand side. Then

If you mentioned relation (1) as a reason, you got the full credit. The complete argument proceeds as follows:

If u is a vector in , then it is a linear combination . Using the relations (1) we can express u as a linear combination of and :

Hence, u is in .

Conversely, If u is a vector in , then it is a linear combination . Using the relations (1) we can express u as a linear combination of and :

Hence, u is in .

2. The columns of any matrix are linearly dependent.

The statement is True. There are more columns than entries in each column.

3. If a set in is linearly dependent, then the set contains more than n vectors.

The statement is False. Counter Example: The set in in linearly dependent but contains only 2 vectors.

4. If the system has a unique solution, then the columns of the matrix A are linearly independent.

The statement is True. Note: If you said that it is a theorem in the text, or proven in class, you got the full credit. The full explanation is: Solutions of the system are coefficients of linear combinations of columns of A which are equal to zero. If there is a unique solution, then the trivial solution must be this unique solution. Hence, if a linear combination of the columns of A is equal to , then all coefficients are 0. This is the definition of linear independence.

5. If the columns of a matrix A are linearly independent, then the system is consistent for every vector in .

The statement is False. Counter Example: Take and .

6. Let A and B be two matrices and C=AB their product. If the third column of B is a linear combination of the first two columns , , then the third column of C is a linear combination of the first two columns , .

The statement is True.
Reason: Use the definition of matrix multiplication. If then

So is a linear combination of and .

7. If is a linearly dependent set in , then the vector is a linear combination of and .

The statement is False. Counter Example:

3.   (10 points) Set up a system of linear equations for finding the electrical currents 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 across a resistance R is related to the (directed) current I by the equation . 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 .

Note: If you used loop currents , , instead of
branch currents then you got the full credit if you wrote:

4. (15 points) 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 to defined by .

Not a linear transformation. (Because of the quadratic term ). It violates the property . Take for example c=2 and . Then T(2(1,1))=T(2,2)=(10,4) which is different from 2T(1,1)=2(5,1)=(10,2). It violates also the additivity the property . Check!

b) T is the map from to defined by .

Linear transformation. Its standard matrix is

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

Linear transformation. Its standard matrix is

5. (15 points) For each of the following sets of vectors in answer both questions:
i) Does the set span the whole of ? and ii) Is it linearly independent? Justify your answer!!! (No credit will be given for an answer without a justification).

1. , ,

The matrix is row equivalent to which is in echelon form.

(i) Yes, the set spans because the matrix has a pivot in every row.

(ii) No, the set is linearly dependent because not every column is a pivot column. (There are more columns than rows).

2. ,

The matrix is row equivalent to

(i) No, the set does not span because there does not exist a pivot in every row (there are less vectors than entries in each vector).

(ii) Yes, the set is linearly independent because there is a pivot in every column.

3. , ,

The matrix is row equivalent to .

(i) No, the set does not span because there does not exist a pivot in every row.

(ii) No, the set is linearly dependent because there does not exist a pivot in every column.

6. (10 points) Let , , and C=AB their product.

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

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

Since , then is a particular solution. If you stopped here you still got the full credit. The row reduced echelon form of the matrix A is The general solution is