Mathematics 317 - Test 3

Date/Time: Friday Dec 5, 12.10-1.00 in Carver 0060

Material is 4.7, Chapter 5 plus 6.1 excluding orthogonal matrices. Questions will be similar to homework questions (see homework page and model solution). Calculators will be allowed, but not necessary. No textbook/notes, but you may have one sheet (2 pages) with theorems/definitions/examples/diagrams, whatever you like. I recommend to have a watch at hand, even if you do not normally wear one (but I will announce the time occasionally).

• Coordinatization, transformation 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)
  • Isomorphisms, 1-1, onto (5.4)
  • Diagonalizable linear operators (5.5)
  • Characteristic polynomial, eigenvalues, eigenvectors, geometric/algebraic multiplicity of an eigenvalue (5.5)
• Dot Product, Orthogonality (6.1)

• 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 transformation is an isomorphism (5.4)
• Find eigenvalues, eigenvectors, characteristic polynomials, multiplicities, possibly applying Theorems 5.23, 5.24, 5.26, 5.27 (5.5)
• 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)

Here is a Practice Exam with model solutions.

Attention. Maybe the test will include a problem about finding the matrix for a given linear transformation with respect to some non-standard basis - see eg examples 5, 6 and 7 in section 5.2. Only that would make this practice exam too long - try to do it in 50 mins! write down how many minutes you took for each question and compare to the marks available.

If you have to perform some row reduction that is really easy, try doing in your head and just write down the end result. The exam will not require hard/long row reductions, but especially if you do not use a calculator, you should be able to do the easy ones fast.