MATH 341 lectures 001&002 spring 2022

See Canvas for information about the textbook, piazza, office hours, exams, and grades. This webpage will collect additional material, and also homework and exam solutions.

The text book

We will follow chapters 1–5 from the book Linear Algebra Done Wrong by Sergei Treil from Brown University. The book is formatted as a textbook from the American Mathematical Society, but the author decided to make it freely available. The particular version we will be using is available here as a free PDF and was downloaded from the author’s website in November 2021.

Exams

Final exam—Sunday May 8, 2:45pm—4:45pm
Note that this time and date differs from the one that was posted in the time table before December 1st.
Location to be announced in your student center sometime at the end of the semester.

Midterm exams — There will be two evening midterm exams on

• Friday February 25th, time: 5:45pm—7:15pm
• Friday April 8, time: 5:45pm—7:15pm

Topics to be covered

1. First topics
   • Sets&the axiomatic method
   • Fields ($\R$, $\mathbb C$, and $\mathbb Q$) and Vector spaces
   • Examples: $\mathbb F^n$, $\mathbb C$ is a vector space over $\R$, $\R$ is a vector space over $\mathbb Q$
   • More examples: the space of polynomials $\mathbb{P}$ and $\mathbb P_n$= polynomials of degree $\leqslant n$
     also: $C([0,1])=$ the space of continuous functions on the interval $[0,1]$
2. Linear combinations
   • spanning sets
   • bases
   • linear independence
   • examples
3. The dimension theorem and a brief review of mathematical induction
   (additional notes to be posted here)
4. Linear transformations
   • geometric examples: rotation, reflection
   • matrix–column multiplication representation of $T:\mathbb{F}^m\to\mathbb{F}^n$
   • the vector space $\mathcal{L}(V, W)$
5. Composition of linear transformations
   • general definition and properties
   • matrix multiplication
   • transposes, $(AB)^\top = B^\top A^\top$
6. Invertible transformations
   • definition & examples
   • $(AB)^{-1}=B^{-1}A^{-1}$
   • isomorphisms
7. Subspaces
   • definition, range, and null space of a linear transformation
8. Solving linear equations (light)
   • row-row-row-reduce: row echelon form
   • multiplication by Elementary Matrices
   • every invertible matrix is a product of Elementary Matrices
9. Determinants
   • defining properties
   • products of elementary matrices; computation by row-reduction
   • $\det(AB)=\det(A)\det(B)$
   • cofactor expansion
   • (optional) sign of a permutation as number of reversals and the definition of $\det A$
   • cofactor formula for $A^{-1}$ & Cramer's rule
10. Spectral Theory.
    • eigenvalues and vectors
    • characteristic polynomial
    • distinct eigenvalues
    • diagonalization
11. Inner product spaces
    • standard inner products and norms in $\R^n$ and $\mathbb C^n$
    • general definition and more examples, $L^2(0, 2\pi)$ and $\mathrm{Tr}(A^\top B)$
    • Orthogonal bases, orthogonal projection and the Gramm—Schmidt procedure
    • Adjoint $A^*=\bar A^\top$
    • Rotations and orthogonal transformations
    • Eigenvalues and vectors of real symmetric matrices: the spectral theorem.

Semester calendar