MATH 416: Abstract Linear Algebra
Spring 2019

Information about exams

There will be three midterms (held outside of class) and a final. We will drop the lowest midterm score.
First midterm: Feb 11, 19:00-20:30, Engineering Hall 106B1
Second midterm: Mar 4, 19:00-20:30, Engineering Hall 106B1 (NB drop date March 8)
Third midterm: April 15, 19:00-20:30, Engineering Hall 106B1
Final exam: May 3, 13:30-16:30, Henry Administration 138

Final Exam

The Final exam will be Friday, May 3, 13:30-16:30, in Henry Administration 138. The final will last all 3 hours and is closed-notes, closed-book, with no calculators allowed.
There will be a conflict exam on Friday, May 3, 19:00-22:00, in 341 AH.

The exam will cover all of the course with emphasis on Chapters 4-5.

A practice final will be available on compass. Solutions to the practice midterm will be posted on compass on May 1.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can be invertible?)
  • inner product
  • The Cauchy-Schwarz inequality
  • The triangle inequality
  • norm
  • orthogonal
  • Frobenius inner product
  • Frobenius norm
  • Gram-Schmidt process
  • orthonormal basis
  • orthogonal projection
  • orthogonal complement
  • operator norm
  • isometry
  • orthogonal matrix
  • unitary matrix
  • QR decomposition
  • singular value decomposition
  • singular values
  • singular vectors
  • adjoint
  • self-adjoint
  • Hermitian
  • symmetric (matrix)
  • normal map or matrix
  • Spectral theorems
  • positive definite matrix

Third Midterm

The third midterm will be Monday, April 15, 19:00-20:30, in Engineering Hall 106B1. The midterm will last all 90 minutes and is closed-notes, closed-book, with no calculators allowed.
There will be a conflict exam on Monday, April 15, 15:30-17:00, in 341 AH.

The exam will cover Chapters 1-3 and Chapter 6 (sections 6.1-6.3).

A practice midterm is available on compass. Solutions to the practice midterm will be posted on compass on April 13.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can be invertible?)
  • linear independence
  • linear dependence
  • finite dimensional
  • infinite dimensional
  • basis
  • standard basis
  • dimension
  • rank (of a linear map or of a matrix)
  • nullity
  • Statement of Rank-Nullity Theorem
  • coordinate representation
  • matrix of a linear map with respect to bases
  • change of basis matrix
  • similar matrices
  • diagonalizable (matrix or map)
  • matrix invariant
  • trace
  • upper triangular matrix
  • algebraically closed field
  • determinant of a matrix
  • determinant of a linear map
  • characteristic polynomial
  • multiplicity of a root
  • multiplicity of an eigenvalue
  • Statement of Cayley-Hamilton theorem

Second Midterm

The second midterm will be Monday, March 4, 19:00-20:30, in Engineering Hall 106B1. The midterm will last all 90 minutes and is closed-notes, closed-book, with no calculators allowed.
There will be a conflict exam on Monday, March 4, 15:30-17:00, in 341 AH.

The exam will cover all the course material covered through February 22; that is, through section 2.6 of the book.

A practice midterm is available on compass. Solutions to the practice midterm will be posted on compass on March 1.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., what kind of thing can be invertible?)
  • isomorphism
  • isomorphic
  • matrix of a linear map
  • product of two matrices
  • transpose
  • conjugate transpose
  • invertible map
  • identity matrix
  • diagonal matrix
  • invertible matrix
  • elementary matrices
  • range
  • column space
  • kernel
  • eigenspace
  • binary linear code

First Midterm

The first midterm will be Monday, February 11, 19:00-20:30, in Engineering Hall 106B1. The midterm will last all 90 minutes and is closed-notes, closed-book, with no calculators allowed.

The exam will cover all the course material covered through February 1; that is, through section 2.1 of the book.

A practice midterm is available on compass. Solutions to the practice midterm will be posted on compass on February 8.

Definitions

You will be asked for complete, precise definitions of about four of the following terms, including context (e.g., span of what?). Remember also that a complete mathematical definition leaves no room for ambiguity: if you give me a definition of what it means for a matrix to be in RREF, I need to be able to use it to decide whether or not any matrix I ever meet is in RREF.
  • linear system of equations
  • solution of a linear system of equations
  • consistent
  • inconsistent
  • unique solution
  • augmented matrix
  • row-echelon form
  • reduced row-echelon form
  • pivot
  • pivot variable
  • free variable
  • underdetermined
  • overdetermined
  • linear combination
  • span
  • standard basis vectors
  • field
  • vector space
  • subspace
  • linear map
  • injective
  • surjective
  • bijective
  • eigenvalue
  • eigenvector