Mathematics 317 - Final
Date/Time: Thursday 12/18, 12-2 in Carver 0060
This will be comprehensive, but the focus is on chapters 3-6.
Calculators will be allowed, but not necessary. No textbook/notes, but you can have all the crib sheest from the previous tests. I recommend to make new ones, though since it helps memorizing the material. I recommend to have a watch at hand, even if you do
not normally wear one (but I will announce the time occasionally).
Keywords are - OK, this will look a little frightening, even if I have weeded out some items. Look at the practice final!
Behind every item is the relevant section of the textbook.
- Determinants
- Connection to row reduction/matrix mult. (3.2,3.3)
- Kitchen Sink Theorem (3.2, table 3.1)
- Expansion by cofactors (3.3)
- Easy evaluation for triangular matrices (3.2)
- Eigenvalues, -vectors, characteristic polynomial, diagonalization (3.4,5.5)
- Vector spaces
- Axioms (4.1)
- Subspace (4.2)
- Span (4.3)
- Linear Independence (4.4)
- Basis, Dimension (4.5)
- Coordinatization, transition matrix (4.7)
- Linear transformations (5.1)
- Matrix of a linear transformation (5.2)
- Diagrams linking transformations, matrices (5.2-5.5)
- Kernel, range, rank (5.3)
- Dimension Theorem (5.3)
- Dot Product, Orthogonality (6.1)
- Simple Properties (6.1)
- Orthogonality and linear independence, orthogonal bases (6.1)
- Orthogonal matrices (6.1)
- Projections (6.2)
Key methods that you should be comfortable with are
- Solve a linear system by row reduction
- Find the rank, determinant, characteristic polynomial, eigenvalues of a
matrix (3.4,5.5)
- Find a basis for a given subspace (eg range/kernel of a matrix)
- Check that a given subset of a vector space is a subspace
- Check that a given transformation is linear (5.1)
- Calculate matrix, kernel, range of a given transformation (5.1, 5.2)
- Apply the Dimension Theorem, possibly find suitable linear
transformation first to solve a given problem (5.3)
- Decide whether a given operator is diagonalizable (5.5)
- Use simple properties of orthogonality, applications to geometry (6.1)
- Find orthonormal/orthogonal basis of a subspace (Gram-Schmidt, 6.1)
Row reduction is not the focus of this exam, but you need it for finding bases for eigenspaces, kernels, ... Together with applications of the Dimension Theorem and the Gram-Schmidt algorithm, I would say this is the core of the methods in the course.
A practice exam with model solutions is on WebCT.
The real exam with a model solution will appear there, too. The grades and scores will appear on WebCT and here, too.