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Sample Midterm Exam
Math 20F Name: 8/22/08 Section:
Read all of the following information before starting the exam:
• READ EACH OF THE PROBLEMS OF THE EXAM CAREFULLY!
- Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct).
- A single 8 1/2 × 11 sheet of notes (double sided) is allowed. No calculators are permitted.
- Circle or otherwise indicate your final answers.
- Please keep your written answers clear, concise and to the poin.
- This test has xxx problems and is worth xxx points. It is your responsibility to make sure that you have all of the pages!
- Turn off cellphones, etc.
- Good luck!
∑^5
1. (0 points) (a) Let A =
, and b =
. Solve Ax = b.
(b) Suppose T
and T
. Find the matrix for the
linear transformation T.
(c) Define linear independence.
(d) Let A =
. Are the columns of A linearly independent?
3. (0 points) (a) Suppose A =
. Compute det(A).
(b) Suppose A =
, and b =
. Use Cramer’s rule to solve Ax = b.
(c) Find the area of the triangle with vertices (2, 3), (4, 7) and (8, 4) using the methods of this course.
4. (0 points) For each statement, mark it true of false. If it is false give a (counter)example or
brief proof. If it is true give a reason - if the reason is a theorem, state the theorem, otherwise give a brief proof. No credit for answers without a correct reason or example. Unless explicitly noted, there are no condition on the dimensions of matrices A and B. (a) If {v 1 , v 2 , v 3 } are linearly dependent, then one of the vectors is a multiple of another.
(b) If Ax = b is consistent for all b, the columns of A span Rm.
(c) If AB = AC, then B = C.
(d) If the transformation T (x) = ABx is onto, then the transformation T ′(x) = Ax is onto
5. (0 points)
(a) If T : Rn^ → Rm^ is a linear transformation verify that the range of T = {v ∈ Rm^ : T (x) = v for some x ∈ Rn}, is a vector space. (Note the range is not necessarily all of Rm. Rm is the co-domain, not the range.).
(b) Verify that the derivative operator (^) dxd is a linear transformation from the vector space of polynomials of degree at most 3 to the vector space of polynomials of degree at most
- Is this linear transformation 1 − 1? Onto?
Scrap Page
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