Math 235 Solution for Midterm 1 Spring 1999
Your grade was calcultated as follows: If your score on question 2 is x and your total score is y then your grade is either y or , whichever is higher. In other words, if your score on question 2 is low, your grade was calculated as if question 2 was worth 20% instead off 35% (and the rest 80% instead off 65%).
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
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 .
The statement is True. There are more columns than entries in each column.
The statement is False. Counter Example: The set in in linearly dependent but contains only 2 vectors.
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.
The statement is False. Counter Example: Take and .
The statement is True.
Reason: Use the definition of matrix multiplication.
If then
So is a linear combination of and .
The statement is False. Counter Example:
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:
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
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).
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.
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.
(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