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Simon Fraser University
Math 232
Midterm 2 Date: 6 July 2007
Instructor : Aaron Bradford Time: 11:30 - 12:
Last Name (print): ________________________ First Name: _______________________
Signature: ______________________________ SFU Email ID: _____________________
Instructions:
1. DO NOT OPEN THIS EXAM UNTIL INSTRUCTED TO DO SO.
- Ensure that you have 5 pages of questions.
- No calculators, notes or books are allowed.
- Except for question 1, credit will not be given for answers with no explanation.
- Answer each question in the space provided. Continue on the back of the previous page if necessary.
- Good luck!
Question Mark Maximum
Total 29
- ( ½ point each ) Mark the following statements as either true or false. No explanation is required.
a. ____
det A + B = det A +det B
b. ____
If all of the entries in A are integers and det A = 1 , then all of the entries in
1
A
−
are
integers.
c. ____
The correspondence
[ ]
x x
B
is called the co-ordinate mapping.
d. ____
2
is a subspace of
3
e. ____ It is possible for the null space of a 10 × 12 matrix A to have dimension 1.
f. ____ The columns of P
C← B
are linearly independent.
- ( 3 points – 1 point ) Suppose that
A
a. Compute the
2,3 - and
3,1 -co-factors of A.
b. The remaining six co-factors of A are
11
C = 2 ,
13
C = − 4 ,
21
C = − 3 ,
22
C = − 2 ,
32
C = − 6 and
33
C = 6. Given that det A = 2 , find
1
A
−
- ( 1 point – 5 points )
a. Let V be a vector space. Define what it means for a set B to be a basis for V.
b. Given
A
, find a basis for each of Col A , Nul A , and Row A.
- ( 2 points – 2 points )
a. Define what it means for a set H to be a subspace of V.
b. Determine if the set
2
a
H a
a
is a subspace of
2
. Justify.
