MIDTERM 1 INSTRUCTIONS:

• Below are some practice problems for the midterm, try to study till you feel comfortable with those
• Midterm 1 covers the first 3 Chapters (3.4 excluded)
• The  midterm will take place in class, usual place and time. Make sure you arrive before 2:20pm. The exam will end at 3:40pm.
• No calculators, books and notes are allowed. If you have your book or notes with you, they should stay in a bag during the exam, not to be seen from outside.

Practice Problems  (The exam will consist in five problems.)  

1. Exercise 1.2-9
2. Exercise 1.2-30
3. Exercise 1.3-32
4. Exercise 1.3-46
5. Exercise 2.2-23
6. Exercise 2.2-30
7. Exercise 2.3-47
8. Exercise 2.3-57
9. Exercise 2.4-31
10. Exercise 2.4-41
11. Exercise 3.1-23
12. Exercise 3.2-46
13. Exercise 3.2-49
14. Exercise 3.3-26
15. Exercise 3.3-31

Checklist for Midterm I

•     Solve a system of linear equations using Gauss-Jordan elimination
•     Reduced row-echelon form (rref) of a matrix, how to find it, and how the rref of a matrix gives the solution of a linear system
•      What is the rank of a matrix?
•      How does the solution of a linear system depend on the ranks of coefficient- and augmented matrices
•      How to add and multiply matrices
•      Matrix multiplication is associative, but not commutative!
•      What is a vector? How to add vectors and take a scalar multiple of a vector? When two vectors are parallel? 
•      What is Rⁿ? What operations can one do with its elements?
•      Calculate the dot product of two vectors in  Rⁿ.
•      What does it mean that two vectors are orthogonal?
•      What is a linear transformation? Can you give some examples?
•      What is the matrix of a linear transformation and how to find it
•      Linear transformations on a plane: scaling, projection, reflection, rotation.
•      What is a composition of linear transformations and how to find its matrix
•      What is the inverse thansformation?
•      Inverse matrix, what is it? how to compute it?
•      What is a subspace of  Rⁿ?
•      What is a linear combination of vectors?
•      What is a span of vectors?
•      Which vectors are said to be linearly dependent?
•      Which vectors are said to be linearly independent?
•      How to test linear dependence/independence?
•      What is the kernel of a linear transformation?
•      What is the image of a linear transformation?
•      Find basis of the kernel and image of a linear transformation.
•      What does it mean that vectors form a basis of a subspace?
•      What is the dimension of a subspace?
•      Can you say "A matrix is invertible" in nine different ways?
•      The rank of a matrix and the dimension of the image, how are they related?
•       If T is a linear transformation then dim(ker(T))+dim(im(T))= ??