From: squash
Date: 4 Mar 1998 22:52:55 -0500
Hi Folks
Alas a topology student came by after class, and I worked with him.
When I got home this evening, I found that the MDCS would not let me log
in! (Rather, it did, let me get some tying done, then Netblazer failed.)
So I've come back to school to type...
Some things to know:
Exam B will cover class material as well as though 2.5 of the text,
*excluding* 1.9.
Def of: Linear transformation (between abstract vectorspaces; what is an
example of a map between vectorspaces that ISN'T linear?
Def of: A map between two SETS being injective, surjective. Examples of maps
that are, and that aren't. Examples of linear maps that are and aren't.
Know how to show that the composition of linear maps is linear.
If S is a linear map, between finite dimensional vectorspaces R^k and
R^n, what is the matrix corresponding to S? If I tell you S(e_1), S(e_2),
..., S(e_k), know how to write down the corresponding matrix.
Conversely, given the matrix B for S, how do you tell from B if S is
injective? surjective?
What is the rotation matrix for rotation of the plane by angle theta
Needless to say [but I'm going to say it anyway], theta is in radians.
What happens when you compose rotations? Know how to derive the
cos and sin of a sum of angles
formulae, the way we did in class (using that the composition of two
rotations is another rotation, etc.).
Given linear transformations T_B and T_A, coming from matrices B and A,
what is the matrix corresponding to the composition
T_B o T_A ?
What is a "shear transformation", and what is its matrix? Example
question. If I rotate the plane by Pi/6, then shear in the horizontal
direction by 3 units, at height one [so the shear leaves e_1 alone, and
sends e_2 to (e_2 + 3e_1)], what is the 2x2 matrix which describes the
resulting transformation? (That is, rotation followed by shear.)
If A is an nxn non-singular matrix, how can I use rref to compute A^{-1}
(that is, A inverse)?
Know how to accurately and completely state the following theorems:
In section 2.2: Thms 6, 7.
In section 2.3: Thms 8,9
Know how to add and multiply partitioned matrices. A typical problem is
P.130: 9, or any of 13-20.
Know how to do an "LU" factorization of a matrix A, under the assumption
that A can be row-reduced to an ref *without* using row-transpositions.
An example is worked out in detail on P.136. Note that L is always
square, and that U is an ref of A. Note that there is NOT a unique LU
decomposition; A can row-reduce to lots of different matrices which are in
ref.
Know how to use an LU factorization of A to solve an equation Ax=b,
where b is a given column vector and x is the unknown.
More forthcoming, (some older stuff)
-J.King