Math 3130 - Linear Algebra - Spring 2016
Exam 2 - Information
Abrams
This exam will last for the entire 75 minute class period of Thursday, April 7, 3:05 - 4:20pm
Reminder of the Calculator Policy: You may use, but will not need a calculator which has no graphing
capability, no internet connectivity, and no QWERTY keyboard.
This exam will cover the material from Chapter 8, all sections, plus some material from Sections 1.8 and 4.6.
You will need to know the appropriate material from the first part of the course (e.g., the definition of a coordinate
vector with respect to some basis.)
I would suggest you read your notes to get a good ’big picture’ of what we have covered. Then review your
homework. For problems you think you are having trouble with, do a few more similar problems in each of the
assigned sections. Only then should you attempt the Practice Exam. Complete the practice exam as if you
were in a real exam setting (closed notes, 75 minutes, etc ...) Compare your answers to those in the Solutions.
Solutions to the Practice Exam (as well as a blank copy of the Practice Exam, in case you’d like to print off an
extra copy) can be found at the course website
http://www.uccs.edu/∼gabrams/CourseMainPageMath3130Spring2016.html
For those problems you get right, you can probably assume that you know the information on that topic
well enough for the exam. For those you got wrong, go to some similar problems from the homework and try to
practice those. If you still need assistance, SEE ME. (Extra office hours listed below.)
Know how to do the following.
If B is a basis of the vector space V and v is a vector in V , find the coordinate matrix [v]B of v with
respect to B.
Give the precise definition of a linear transformation. (p. 447)
Prove that a given function is between vector spaces is or is not a linear transformation. The function
may be given as a formula, or described geometrically.
If T : V → W is a linear transformation, and you are told the values of T on a basis for V , be able to
compute the value of T for any given input vector from V .
Know the definition of the kernel and range of a linear transformation. Be able to describe these for
specific transformations (either by describing the vectors explicitly, or by producing a basis for these subspaces.)
Know the definition of the nullity and rank of a linear transformation. Know the statement of the
Dimension Theorem for linear transformations (p 455).
Be able to fill in the blanks appropriately:
”Let V be an n-dimensional vector space with basis B = {u1 , u2 , ..., un } and let W be an mdimensional vector space with basis B 0 = {w1 , w2 , ..., wm }. Let T : V → W be any linear transformation.
Let A denote the matrix
..
..
..
(fill in the blank with ...)
[T (u1 )]B 0 . [T (u2 )]B 0 . · · · . [T (un )]B 0
Then for every vector v in V,
A[v]B = (fill in the blank with [T (v)]B 0 )
OR T (v) = (fill in the blank with GB 0 (A[v]B ). )
The matrix in the previous paragraph is called the matrix of T with respect to the bases B and B 0 . Be
able to compute this matrix for a given transformation T and bases B, B 0 . Then be able to compute T (v) directly,
AND by performing the computation indicated in the previous paragraph.
(over)
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If B and C are two bases for the same vector space V , be able to compute the transition matrix from C
to B. (This is the matrix we call PC→B . It is sometimes just denoted by P .)
Given [v]C find [v]B by computing the matrix product P [v]C .
Know that the transition matrix from B to C is the inverse of the transition matrix from C to B.
Know the definition of similar matrices.
If T : V → V is a linear operator on V , and B, C are two bases for V , be able to: Find the matrix [T ]B
of T with respect to B, B; Find the matrix [T ]C of T with respect to C, C; then know how these are related:
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[T ]C = PB→C [T ]B PC→B = PC→B
[T ]B PC→B .
Office hours and other help between now and the exam date: (unless otherwise indicated, all office
hours are held in my office, Engineering Building Room 288):
Standard office hours:
Thursday March 31, 11:45 - 12:15 and 4:30 - 5:00,
Tuesday April 5, 11:15 - 11:45 and 4:30 - 5:00
Thursday April 7, 11:45 - 12:15
and by appointment.
Extra office hours:
Wednesday April 6 I will be in my office 3:00 - 3:30
Email messages to abrams@math.uccs.edu are encouraged!
Also, remember:
- the Supplemental Instruction sessions with Ikko Saito: Tuesdays and Thursdays, 12:20 - 1:50pm, Columbine
Hall Room 105 and
- The Math Center, located in ENGR Room 233. Free drop-in tutoring. Hours for this semester are: Monday
- Thursday 8 am - 7 pm; Friday 8 am - 4:30 pm; Sunday 11 am - 3 pm.
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