No aids of any kind: no calculators, no notes, no books.
No cell phones, no electronic devices of any kind.
No make-up exams will be given.
Midterm Exam I. Wednesday, February 5.
Covers Sections 0 to III (except 2.6) from the syllabus.
You will be expected to know (and be able to recite) the
following:
Definitions.
Augmented coefficient matrix of a linear system of equations
Row equivalent matrices
Matrix in row echelon form
Matrix in reduced fow echelon form
pivots, pivot columns, pivot variables
Injective, surjective, bijective maps
Image and preimage of subsets of domain/codomain
Complex numbers
Field
Characteristic of a field
Vector space
Subspace
linear (in)dependence
Spanning families
Bases
Canonical basis of $\mathbb{F}^n$.
Dimension
Sum and intersection of subspaces
Facts.
Theorem about existence and uniqueness of the RREF of a matrix
Basis extension theorem
Exchange lemma (either the version in the book, or the version from
class)
Also, be able to perform the following:
Algorithms.
Row reduction algorithm or Gauss-Jordan elimination
Use the Gaussian elimination procedure to put a matrix into REF
Apply the back substitution procedure to put a matrix in REF into
RREF
Determine if two matrices are row equivalent using row reduction
Determine the general solution of a system of linear equations using
row reduction
Select a basis for a subspace of $\mathbb{F}^n$ from a spanning family
by applying the row reduction algorithm
Implement the base extension theorem in $\mathbb{F}^n$ using row reduction
Determine if a given family of vectors in $\mathbb{F}^n$ is is
linearly independent or not, using the row reduction algorithm
Problems.
Solve simple "word problems", which lead to systems of linear equations
Determine if a system of linear equations is consistent or not
Determine if a given set with operations is a vector space
Determine if a subset is a subspace
Find a basis of a given vector space
Practice Exam. This practice exam is of
course much longer than the actual midterm will be.
Please discuss questions or solutions on Piazza, there is no answer
key to this practice test.
Your percentage score was computed as
$(2\times\#\text{points}+50)$.
The class average (of the adjusted scores) was 91.5%.
Midterm Exam II. Monday, March 16.
Covers Sections 0 to IX from
the syllabus (Except 2.6, 4.5, 5.6, VIII, 9.4), with somewhat more emphasis
on Sections IV-IX.
In addition to the material listed above for Midterm I, you will be
expected to know (and be able to recite) the
following:
Definitions.
Determinant of a matrix,
Determinant of an endomorphism,
Linear map
Kernel and image of a linear map
Mono-, epi-, iso-, endo-, automorphism
Rank of a linear map
The matrix $[f]^\mathcal{B}_\mathcal{C}$ associated to a linear map
$f$, and bases $\mathcal{B}$ and $\mathcal{C}$
Formula for the entries of a product of two matrices (p. 87)
Formula for the matrix of a composition
$[fg]^\mathcal{A}_\mathcal{C}
=[f]^\mathcal{B}_\mathcal{C}\,[g]^\mathcal{A}_\mathcal{B}$ (p. 87)
Rank of a matrix, row rank and column rank
Eigenvalue/Eigenvector of an endomorphism,
Eigenvalue/Eigenvector of a matrix,
Diagonalizable matrix/endomorphism,
Eigenspace,
Geometric multiplicity of an eigenvalue,
Characteristic polynomial of a matrix/an endomorphism,
Algebraic multiplicity of an eigenvalue,
Facts.
Unique existence of the determinant,
Behaviour of determinant with respect to row/column operations,
Laplace expansion along a row or column,
Determinant of transpose,
Significance of determinant for invertibility,
Formula for determinant and inverse of a $2\times2$-matrix,
Determinant of product and inverse,
Rank criterion for solvability of an (in)homogeneous system of equations,
Structure of the solution set of an (in)homogeneous system of equations,
Dimension formula for sums of subspaces
Dimension formula relating image and kernel of a linear map
The universal mapping property of bases (Chapter 4, Remark 2)
Equality of row and column rank.
Linear independence of eigenvectors for different eigenvalues,
Criterion for diagonalizability in terms of geometric multiplicities of eigenvalues,
Eigenvalues = zeros of the characteristic polynomial,
Geometric multiplicity at most as large as algebraic multiplicity,
Also, be able to perform the following:
Algorithms.
Find a basis of the kernel of a linear transformation by putting the
coefficient matrix into RREF,
Find a basis of the image of a linear transformation by putting the
coefficient matrix into REF,
Find a basis of the row space of a matrix by putting the
coefficient matrix into REF,
Find the rank of a matrix using elementary row operations
Find the inverse of a matrix using elementary row operations
Problems.
Find the determinant of any square matrix using row/column operations
and row/column expansions.
Determine if a map is linear or not
Find the matrix of a linear map with respect to given bases of domain
and codomain
Find the matrix of an endomorphism with respect to a basis.
Diagonalize a matrix (if possible), i.e., write $A=PDP^{-1}$,
Find the general solution (with undetermined coefficients) of a discrete dynamical system (if the transition matrix is diagonalizable),
Solve a discrete initial value problem,
Analyze long-term behaviour of discrete dynamical systems,
Practice Exam. This practice exam is of
course much longer than the actual midterm will be. Skip Problems 7,
8, 15b, 16, 17, 18. Replace by Exercises 1.1, 1.2, 1.3, 1.5, 1.9
from the notes on dynamical systems.
Your actual exam with solutions.
To compute your percentage score simply add 65 to your point
score.
(You get the percentage score that I would have counted, if this
midterm were counting.)
The class average (of the adjusted scores) was 88%.
Final Exam. April 24, 12:00-15:00.
Covers the entire syllabus. In addition
to the material listed above for the midterms, you will be tested on
the following:
Definitions.
Inner product,
Euclidean vector space,
Standard inner product for $\mathbb{R}^n$,
Norm,
Angle,
Orthogonal vectors,
Othonormal system of vectors,
Orthogonal complement of a subspace,
Orthogonal map,
Orthogonal matrix.
Group,
Subgroup,
Abelian group,
Self-adjoint endomorphism,
Symmetric matrix,
Equivalence relation,
Equivalence class,
Quadratic form,
Rank of a quadratic form,
Signature of a real quadratic form.
Facts.
Formula for rotation matrices in $\mathbb{R}^2$.
Cauchy-Schwarz inequality,
Properties of the norm,
Expansion with respect to an orthonormal system,
Formula for the projection (given an ON basis of the subspace),
Gram-Schmidt orthonormalization process (including uniqueness statement),
How to recognize orthogonal maps by images of ON bases,
Equivalent ways to recognize an orthogonal matrix,
Properties of matrices of rotations and reflections in
$\mathbb{R}^2$.
Basic examples of groups,
Fundamental theorem of algebra (without proof),
Orthogonality of eigenvectors of a self-adjoint endomorphism,
Invariance of orthogonal complement of an eigenvector of a self-adjoint endomorphism,
Relationship between self-adjoint endomorphisms, orthonormal bases, and symmetric matrices,
The Spectral Theorem/Principal Axes Theorem (existence of orthonormal
basis of eigenvectors),
Spectral decomposition of self-adjoint operators,
The Rank Theorem (classification of matrices up to row/column equivalence),
Jordan Normal Form theorem (classification of complex square matrices
up to similarity) (without proof),
Classification of real symmetric matrices up to orthogonal similarity,
Existence of Sylvester Basis for a real quadratic form,
Classification of real symmetric matrices up to coordinate equivalence
in terms of Sylvester Normal Form.
Algorithms.
Apply the Gram-Schmidt process to a given independent family of vectors.
Find the matrix of a quadratic form,
Diagonalize a quadratic form, i.e., write $A=Q^t D Q$,
Find a Sylvester Basis of a quadratic form.
Problems.
Find the powers of a matrix,
Orthogonally diagonalize symmetric matrices,
Find the principal axes of conic sections $q(x)=1$, and
sketch.