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