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: and are free variables.

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

Answer: The question is equivalent to:

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

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 and (indeed, ).

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

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

(i.e., such that ).

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 and are free. We get

The two vectors on the right span the plane P.

3. a) Is the set ,

i) (5 points) linearly independent?

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

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

Answer: No, the two vectors do not span . 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 are always linearly dependent.

ii) (5 points) span ?

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

4.   (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.

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 , , , 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).

, , , , , .

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 , , be three vectors in . Then the set is linearly dependent (regardless of the choice of , , ).

Answer: TRUE. Let , , and . The three vectors satisfy the linear relation

2. The columns of any matrix are linearly dependent.

Answer: FALSE. Counter Example:

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

Answer: TRUE. Since the columns span , 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 to defined by .

Answer: A linear transformation with standard matrix

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

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

(1) , for any two vectors in , and

(2) , for any scalar c and vecotr .

It fails to satisfy both (1) and (2). Take for example and . 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 and c=2.

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

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

Hence, , while

The standard matrix is