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Simon Fraser University
Math 232
Midterm 1 Date: 8 June 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.
- You may not use determinants (or any other material beyond chapter 2) to answer any
questions.
- Good luck!
Question Mark Maximum
Total 26
- ( ½ point each ) Mark the following statements as either true or false. No explanation is
required.
a. ____ (^) The weights 1
p
c … c in the linear combination 1 1 p p
c v + + c v
… cannot all be
zero.
b. ____ If the equation Ax = b
is inconsistent, then b
is not in the set spanned by the
columns of A.
c. ____ The equation Ax = b
is homogeneous if the zero vector is a solution.
d. ____ If x
and y
are linearly independent, and if z ∈ Span x y ( , )
, then { x y z , , }
is
linearly dependent.
e. ____
If A is a 3 × 5 matrix and T is a transformation defined by T ( x )= Ax
, then the
domain of T is
3
.
f. ____ If A can be row-reduced to the identity matrix, then A must be invertible.
- ( 3 points ) Compute AB , where
A
and
B
. Why is BA undefined?
- ( 2 points – 3 points )
a. Without performing any row-operations, decide whether or not the matrix
is invertible. Justify your answer.
b. Find the inverse of the matrix
A
. Show your work.
- ( 1 point – 2 points – 2 points )
a. Define what it means for a transformation :
n m
T → to be linear.
b. Show that the transformation
2 2
T : → given by ( ) ( )
1 2 1 2 1 2
T x , x = 3 x − 5 x , x + 2 x is
linear by verifying that it satisfies the definition of linear.
c. Find the standard matrix of the transformation T given in (b)
