Exam 3 Info
MCS 221 Exam 3 Info
Date & time: Thursday, October 31, 1:30-2:20
Place: Classroom
Coverage: Penney chapter 3, class notes:
Linear Transformations
Test 3 will be a closed-book, closed-notes examination.
You may use one new handwritten 3"-by-5" note card.
You may also use your Exam 1 and 2 note cards.
You may not use a calculator.
The problems will be rigged so that you should be able to the calculations
by hand.
There will not be any Maple problems on the test.
Be prepared to do problems and use knowledge from the following areas.
- 3.1: Linear transformations
- Definition of a linear transformation,
checking/proving a given transformation is linear
- Rotations
- Images of figures
such as line segments, polygons, and circles
under linear transformations
- The Matrix Representation Theorem
- Examples of linear transformations that do not have
matrix representations
- 3.2: Matrix multiplication (composition)
- Matrix multiplication is defined so that a product of
matrices corresponds to a composition of transformations.
- Penney's definition of matrix multiplication
in terms of the already-defined product of a matrix and a
column vector, definition (M), p. 153
- Alternative ways of computing or viewing a matrix product
- Properties of matrix multiplication
- Associative law
- Left distributive law
- Right distributive law
- Scalar law
- Not a law: commutativity
- (AB)t =
Bt At
and (At)t = A.
- 3.3: The image of a transformation
- The image of a set under a transformation,
the image of a transformation (= its range)
- The preimage, or inverse image, of a set
- Problem: determine a basis for the image.
- Proof problem:
Prove that an image or preimage is
or is not a vector (sub)space.
- Theorem: For an m-by-n matrix transformation,
dim(image) = n - dim(nullspace).
(This is the rank-nullity theorem restated.)
- Many-to-one, one-to-one, onto, invertible
- The Inverse Theorem: conditions equivalent to invertibility
of a matrix transformation
- Rank of Products Theorem
- 3.4: Inverses
- The general notion of an inverse transformation
- Algorithm for determining the inverse of a matrix
- Properties of the inverse A-1 of a matrix
A: A-1 is unique;
A A-1 = I;
A-1 A = I.
- (AB)-1 =
B-1 A-1
and (A-1)-1 = A.
- 3.5: The LU factorization
- The LU-decomposition Theorem
- Algorithm for finding the L and U factors
when no row swaps are needed in Gaussian elimination
- Solving A x = b when A = L U
by forward substitution and back substitution