# Classes

## Chapter 2:

#### Monday December 1 (slides)

We finished our discussion of complex numbers. Solving equations, trigonoometric form, and De Moivre's Theorem.#### Friday November 28 (slides)

The basics of complex numbers; how to write complex numbers, and how to add, subtract, multiply, and divide them. Plotting complex numbers in the complex plane. Conjugate and modulus.#### Wednesday November 26 (slides)

Finished up with eigenvalues and eigenvectors, and moved on to complex numbers (in both Goodaire and Poole, there is an appendix on complex numbers).#### Monday November 24 (slides)

§3.4. More examples of eigenvalues and eigenvectors for \(3\times3\) matrices.#### Friday November 21 (slides)

§3.4, computing eigenvalues and eigenvectors. This is a moderately lengthy step-by-step process that relies on both our ability to compute determinants and to solve homogeneous systems through row-reduction.#### Wednesday November 19 (slides)

§3.4. Eigenvalues and eigenvectors. We saw eigenvectors as vectors that get transformed (by a matrix) to scalar multiple of themselves, and even if this doesn't seem interesting, it has a whole host of applications (like search engines and image processing).#### Monday November 17 (slides)

§3.1. Computing determinants of larger square matrices.#### Friday November 14 (slides)

§3.2 (yes, we've skipped §3.1 for now: we'll do it Monday). Properties of determinants, and lots of \(2\times 2\) examples. This video and this one might be helpful.#### Friday November 7 (slides)

Practice computing inverses using row reduction, §2.9. (If you're interested, you could set up a \(2\times 2\) matrix (with entries \(a,b,c,d\)) and see that the row-reduction technique gives the correct inverse.) Only square matrices can have an inverse, and not all square matrices do.#### Wednesday November 5 (slides)

Computing inverses of matrices using row-reduction techniques. §2.9.#### Monday November 3 (slides)

Which system would you rather solve: \[ \begin{align*} x_1 + 2x_2 + 3x_3 &= 5\\ x_1 + x_2 + x_3 &= 2 \\ 2x_1 + 2x_2 + 3x_3 &= 6 \end{align*} \] or \[ \begin{align*} x_1 + 2x_2 + 3x_3 &= 5\\ - x_2 - 2x_3 &= -3 \\ x_3 &= 2 \end{align*} \] or \[ \begin{align*} y_1 &= 5\\ y_1 + y_2 &= 2 \\ 2y_1 + 2y_2 + y_3 &= 6 \end{align*} \]#### Friday October 31 (slides)

§2.6, Homogeneous Systems, in particular, we focus on the material at the end of the section concerning linear independence.#### Wednesday October 29 (slides)

We finished solving \[ \begin{align*} x - y - z +2w &=1 \\ 2x - 2y - z + 3w &= 3 \\ -x + y - z &=-3 \end{align*} \] and did an example concerning linear combinations. This solving a system video might be helpful.#### Monday October 27 (slides) and (visuals)

We used row reduction to solve three systems (we'll finish another on Wednesday). We also plotted the planes and observed the intersections: have a look \[ \begin{align*} x + 2y - z &= 3 \\ 2x + 3y + z &= 1 \end{align*} \] and \[ \begin{align*} x - y + z &= 2 \\ 2x + y -6z &= 7\\ 6x-10z &=5 \end{align*} \]#### Friday October 24 (slides)

Elementary row operations, pivots, and row echelon form. Several examples. This Row reduction video might help.#### Wednesday October 22 (slides)

§2.4. Solving systems of equations. Writing a system of equations as an augmented matrix and vice versa. Manipulating systems (and augmented matrices) in such a way that the solution remains the same.#### Monday October 20 (slides)

§2.3. Computing inverses of \(2\times 2\) matrices. A 2x2 matrix is invertible if and only if its determinant (\(ad-bc\)) is not zero.#### Friday October 17 (slides)

Went over question 2 of test 1. Can you find a vector u so that Cu = [2, 1]? This is equivalent to solving a system of two equations. We did so, and I presented a matrix that, when multiplied by C, gave the identity matrix. This paves the way for matrix inverses.#### Wednesday October 15 (slides)

§2.3, matrix transpose. A vector is a one-column matrix. Multiplication of a matrix and a vector. We had three vectors and three matrices, and explored what happened to the three vectors when they were multiplied by each of the matrices. We finished with the question, Can you find a vector u so that Cu = [2, 1]?#### Wednesday October 8 (slides)

§2.1, matrix multiplication examples. Square matrices, the identity matrix.#### Monday October 6 (slides)

§2.1, matrices. Matrix multiplication as dot product of rows with columns. Matrix addition, subtraction, scalar multiplication (briefly). To come: more matrix multiplication examples.## Chapter 1: Euclidean n-space

#### Friday October 3 (slides)

§1.5, Linear Independence.

#### Wednesday October 1 (slides)

§1.4: Projections, and how we can use projections to find the distance from a point to a plane.

#### Monday September 29 (slides)

§1.4: Projections. This worksheet illustrates the projection of vectors in 3D.

#### Friday September 26 (slides)

We explore two parametric lines, and ask whether they intersect. Regarding the line of intersection of planes, this Plane intersection worksheet will help with visualizations, and this video may be helpful. Regarding the distance from a point to a plane, this video concerns similar ideas, and may be helpful.