# FP3: matrices and vectors

Resource for all this:

Notes on things done differently from book, or not covered in book, and some proofs of things left unproved in the book

Abridged version of notes

“Don’t do as the books says” notes on vectors

3×3 DETERMINANTS

Starter: Reprise 2×2 determinants

180123matrices22recap

Algorithm: 3×3 determinants

Activity: Ex.6B

“Diagonals” method of calculating determinants

INVERSES AND TRANSPOSES OF 3×3 MATRICES

Starter: matrix×matrix of cofactors

Exposition: transposes
Exposition: inverting matrices

Activity: Ex.6C

Homework: Ex.6C

Exercise on determinant rules

Exercises on determinant rules

3×3 MATRICES AS LINEAR TRANSFORMATIONS IN 3D

Starter: Reprise 2×2 matrices as linear transformations in 2D

Exposition: Equations of planes and lines in 3D

Equations of lines: write down the typical point for each of the lines in Ex.5D Q.1

Note that there are lots of different valid answers

Equations of planes: write down the typical point for each of the planes in Ex.5E Q.3

Note that there are lots of different valid answers

Exposition: Transformations in 3D. Look at Examples 17 and 18.

Look at Example 16 p.157 and do Ex.6D Q.4. Look at Example 17 p.157 and do Ex.6D Q.7.

Look at Example 18 p.158 and show that the image is a line without doing more working than the bottom of p.158.

Ignore the nonsense on p.159 of the textbook, which includes dividing by zero.

Look at Example 21 on p.163. Do Ex.6E Q.7. Do Ex.6E Q.1

Activity: Ex.6D, 6E

LESSON 3: EIGENVALUES AND EIGENVECTORS

Summary sheet on eigenvalues, eigenvectors, and diagonalisation

Introducing eigenvalues (docx)

Introducing eigenvalues (pdf)

Geogebra: visualising eigenvectors

Eigenvalues summary, and diagonalisation (docx)

Eigenvalues summary, and diagonalisation (pdf)

Shortcuts for calculating eigenvalues and eigenvectors

How the formulas for An, for a matrix A, which you prove by induction in FP1, can be got by diagonalisation

Ax = λx

det (A−λI) = 0

Activity: Ex.6F Q.1-8

Exposition: Eigenvalues in engineering

Resource:

Discussion: what if complex eigenvalues?

DIAGONALISING MATRICES

In FP1 we found out what (some) matrices looked like by plotting where they mapped the points (1,0) and (0,1) to.

Here a computer has taken a matrix and shows where it maps a whole heap of other points on the unit circle (as well as the points (1,0) and (0,1): the red line is what the line-segment from (0,0) to (1,0) maps to, and the blue line is what the line-segment from (0,0) to (0,1) maps to).

Now airbrush the axes out of the picture

and draw in new axes along the lines you can see in the picture (lines in the direction of the eigenvectors) where the matrix just maps every point on the line further out (in this case; or could be closer in, for an another matrix) along the same line.

The geometrical transformation is still the same, but with different axes the matrix (table of numbers) representing it will be different.

With the new axes, the matrix is just a scaling along the two axes (no shearing).

It is a diagonal matrix: every element in it is zero except the leading diagonal (and the elements there are the eigenvalues).

In fact, it is just this (if we swivel the picture so that the axes are straight across and straight up).

We have simplified both the description of the geometrical transformation (to just a scaling, possibly by different amounts, in two perpendicular directions), and the matrix for it (to a diagonal matrix).

For a matrix A, the new axes along the eigenvector directions can be described by a matrix P, where P is a matrix formed by putting the normalised eigenvectors for A, one by one, into the columns of P.

The change of axes changes the table of numbers (matrix) describing the same transformation to P−1AP

P−1AP = D, which is a diagonal matrix consisting of the eigenvalues of A down the leading diagonal, and zeroes everywhere else.

The normalised eigenvectors can be in any order in P, and the eigenvalues in any order in D, but the eigenvectors and eigenvalues have to be in the same order (first eigenvalue is the eigenvalue corresponding to first eigenvectors, second eigenvalue is the eigenvalue corresponding to the second eigenvector, and so on.

This calculation is especially useful and easy if A is a symmetric matrix. Then, amazingly, all the eigenvalues are real, and the eigenvectors are all perpendicular (orthogonal) to each other. PTP=I (if you’re asked to verify that eigenvectors are orthogonal, you do it by verifying PTP=I), and so P−1=PT. In other words the inverse is very easy to calculate.

PTAP = D

This special case of symmetric matrices is important in real life, and in FP3 you deal with diagonalisation only for symmetric matrices.

Activity: Ex.6G

Homework: Ex.6G Q.2, 3, 6, 9, 10. Review Questions p.196 Q.24. Ex.6F Q.2(a), 3, 4

LESSON 4: DOT PRODUCT AND CROSS PRODUCT

Formulas for dot and cross product: summary

Cross product in motor effect

Calculating cross product

Homework: Ex.5A Q.1-6, Ex.5B Q.1-6

NOTE ON QUATERNIONS

Dot product and cross product come from the multiplication of four-dimensional numbers (quaternions). The product of two all-“imaginary” quaternions

(a1i+b1j+c1k)(a2i+b2j+c2k) = −a.b + a×b

This explains why dot product and cross product “work” fairly well despite lacking many features of “proper” multiplication operations, and why, despite being completely different, they have some subtle links.

Quaternions can also be represented as 2×2 matrices of complex numbers.

Resource: Videos on quaternions

AREAS, VOLUMES, AND TRIPLE PRODUCT

Areas, volumes, uses of vectors in 3-D geometry (docx)

Areas, volumes, uses of vectors in 3-D geometry (pdf)

Cross product as area of parallelogram
Triple product as volume of parallelepiped. Volume of tetrahedron.

Classwork and homework: Ex.5B Q.6, 7, 8, 9, 12 (areas)

Ex.5C Q.1, 2, 3, 5, 6 (volumes)

LESSON 5-6: USES OF VECTORS IN 3D GEOMETRY

Starter: reprise equations of lines and planes in 3D

Ex.5D Q.1, 2, 3 (equations of lines)
Ex.5E Q.1, 3, 4 (equations of planes)

Resources: Notes on methods different from the book

Video on different method for distance from a point to a line

Video on different method for line of intersection of two planes

Exercises: see list below, or download this pdf

Line of intersection of two planes: Ex.5F Q.3, Ex.5G Q.9 (a), (b)

Angle between planes: Ex.5F Q.4,5; Ex.5G Q.11. Angle between lines Ex.5G Q.16

Angle between line and plane: Ex.5F Q.8; Ex.5G Q.12

Distance from point to plane: Ex.5F Q.9, Ex.5G Q.6, Q.13 (distance between a line parallel to a plane, and the plane, is the same as the distance from any point – might as well be the “a” point – on the line to that plane); Review Exercises 2 Q.16 p.195

Distance between parallel planes: Ex.5F Q.10

Distance between skew lines: Ex.5F Q.11, Ex.5G Q.1, 2, 3

Distance from point to line, and distance between parallel lines: Ex.5F Q.12; Review Exercises 2 Q.19 p.196