##
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