Midterm I: Wednesday Feb 22, 2.30-3.50pm, Earth&Space 069

Midterm I will cover Chapters 1 and 2 and Section 3.1 in the textbook. You should be able to:
    1) solve linear systems,
    2) find the reduced row echelon form of a matrix,
    3) determine the rank of a matrix,
    4) determine when a transformation is linear,
    5) write the matrix of a linear transformation,
    6) calculate the product of two matrices,
    7) find the inverse of a matrix,
    8) determine the image and kernel of a linear transformation.

To prepare well for the exam review your notes, textbook and homework. You should also go through the following review exercises at the end of each chapter:
Chapter 1:         1-3, 5-9, 11-14, 16-20, 23, 34
Chapter 2:         3-13, 16-18, 20-25, 27, 30

Midterm II: Wednesday March 29, 2.30-3.50pm, Earth&Space 069

The exam will cover the materials of Chapters 3 and 4 in the textbook. You should be able to:

Find a basis for the image and kernel of a matrix (§3.2-3.3)
Calculate the dimension of a subspace of R^n and the rank and nullity of a matrix (§3.2-3.3)
Find a basis of R^n “adapted” to a linear transformation, that is, a basis such that the matrix that represents the linear transformation is as simple as possible (§3.4)
Verify that a set is a vector space, find a basis and calculate its dimension (§4.1)
Calculate kernel and image of a linear transformation between vector spaces (§4.2)
Given bases, calculate the matrix that represents a linear transformation between vector spaces (§4.3)  

Final exam: Wednesday May 10, 8-10.45am, Old Engineering 143

The exam will cover all that we have done during the semester, with an emphasis on the last part of the course. Look at the Schedule & Homework section to see which sections in the textbook we have covered and which sections you can skip.

Things you should be familiar with to do well on the exam:

solve a linear system using the Gauss-Jordan elimination process
establish when vectors are linearly independent and when a vector is a linear combination of others
calculate rank and nullity of a matrix and find bases for its image and kernel
understand linear transformations defined using geometry (projections, reflections, rotations) 
calculate kernel and image of a linear transformation between vector spaces
calculate the matrix of a linear transformation between vector spaces with respect to a given basis
apply the Gram-Schmidt process to find orthonormal bases
use orthonormal bases to calculate projections onto subspaces
manipulate algebraically the operation of transpose and inverse of a matrix
calculate the determinant of a square matrix to decide whether the matrix is invertible
solve a linear system using Cramer’s Rule
find eigenvalues and eigenvectors and diagonalize a matrix

To prepare for the exam review past homework assignments, your notes, the textbook and do plenty of exercises (there are lots of them in the textbook!). Here’s a possible short selection of exercises to review all that we have done during the semester:

- 5 or 7 in §1.2
- 11 in §2.2
9 or 11 in §3.3
one of 7, 9, 13, 15 in §3.4
25 or 27 in §3.4
51 or 53 in § 4.2
7 or 9 in § 4.3
21 or 23 in §4.3
27 in §5.1
7 or 13 in §5.2
one of 5, 7, 21, 23 in §5.3
15 or 17 in §6.1
23 in §6.3
one of 15, 17, 19 in §7.1
one of 3, 5, 7, 11 in §7.3
45 in §7.3

I will hold office hours as follows:

Monday May 8, 4-5pm, Math Tower 2-121

Tuesday May 9, 11am-1pm, Math Tower 2-121
                           1.30-2.30pm in MLC

 If you need help outside of these times, write me an email and we will arrange a time to meet.

Exam Rules:

Calculators are not needed and therefore not allowed.

All electronic devices (except watches) must be turned off. In particular, cellphones are not allowed. If you take your cellphone out for any reason (even just to check the time), you will be asked to turn in your exam paper and to leave the room.

Notes, textbooks, etc. are not allowed. Only the test paper and pens/pencil/eraser should be on your desk.

No consultations with others. Please raise your hand if you have any question.