Math 235 Midterm 1 solution Fall 2000
Answer: The row reduction takes four steps:
b) Find the general solution for the system.
Answer: and are free variables.
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.
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 ?
Answer: Row reduce.
If , then we have a pivot in every row (and the four vectors span ).
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).
, , , , , .
Answer: TRUE. Let , , and . The three vectors satisfy the linear relation
Answer: FALSE. Counter Example:
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.
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