Math 260, Fall 2018, Assignment 3

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

Carefully define the following terms, then give one example and one non-example of each:

1. $n\times m$ matrix.
2. $n$-component column vector.
3. $\mathbb{R}^n$.
4. $n$-component row vector.
5. $\left(\mathbb{R}^n\right)^*$.
6. Geometric vector.
7. Addition (of matrices, or of column vectors).
8. Addition (of geometric vectors).
9. Scalar multiplication (of matrices, or of column vectors).
10. Scalar multiplication (of geometric vectors).
11. Dot product (of two column vectors).
12. Length (of a column vector).
13. Angle (between two non-zero column vectors).

Carefully describe the following algorithms:

1. Procedure to write all solutions of a linear system, once it is in reduced row-echelon form.

Carefully state the following theorems (you do not need to prove them):

1. Numerical instability of Gauss-Jordan elimination. (This is not really a theorem, but please be able to give an example of an alarming loss of precision when this algorithm is used to solve a system of linear equations.)

Solve the following problems:

1. Section 1.3, problems 10, 11, 12, 27, and 28.
2. Section 5.1, problems 1, 3, 5, 6, 7, 8, 9, and 10.
3. In ordinary three-dimensional space, is it possible for two planes to intersect in exactly one point? Give a rigorous algebraic proof of your answer. (Hint: each plane is represented by an equation, so their intersection is the solution set of a $2\times 3$ system of linear equations (whose augmented matrix is thus a $2\times 4$ matrix), and this is equivalent to some system in reduced row-echelon form. Now think about all of the possible "types" (as defined in exercise 1.2.22, last week) of $2\times 4$ matrices.)

(N.B.: After we have studied the theory of dimension in chapter 3, we will see that it is possible for a pair of planes in $\mathbb{R}^4$ to intersect in exactly one point.)

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Solutions

Definitions:

1. An $n\times m$ matrix is a rectangular array of numbers, having $n$ rows and $m$ columns and written within brackets, as follows: $$\begin{bmatrix}a_{1,1}&\dots&a_{1,m}\\\vdots&&\vdots\\a_{n,1}&\dots&a_{n,m}\end{bmatrix}.$$
2. An $n$-component column vector is an $n\times1$ matrix.
3. $\mathbb{R}^n$ is the set of all $n$-component column vectors.
4. An $n$-component row vector is a $1\times n$ matrix.
5. $\left(\mathbb{R}^n\right)^*$ is the set of all $n$-component row vectors.
6. A geometric vector is a directed line segment (having an initial point and a terminal point). Two geometric vectors are regarded as identical if one can be translated onto the other.
7. The sum of two $n\times m$ matrices is the $n\times m$ matrix obtained by adding corresponding entries of the given matrices.
8. The sum of two geometric vectors, say $\vec{v}$ and $\vec{w}$, is the vector whose initial point is the initial point of $\vec{v}$ and whose terminal point is the terminal point of $\vec{w}$, provided that the vectors have been translated so that the terminal point of $\vec{v}$ coincides with the initial point of $\vec{w}$.
9. If $M$ is a matrix and $k$ is a scalar, then $kM$ is the matrix obtained by multiplying every entry of $M$ by $k$.
10. If $\vec{v}$ is a geometric vector and $k$ is a non-negative scalar, then $k\vec{v}$ is the vector having the same direction as $\vec{v}$ but whose length is $k$ times the length of $\vec{v}$. If $k$ is a negative scalar, then $k\vec{v}$ is the vector having opposite direction to $\vec{v}$ but length $\left\lvert k\right\rvert$ times the length of $\vec{v}$.
11. The dot product of $\vec{x}=\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}$ and $\vec{y}=\begin{bmatrix}y_1\\\vdots\\y_n\end{bmatrix}$ is $\vec{x}\cdot\vec{y}=x_1y_1+\dots+x_ny_n$.
12. The length of $\vec{x}$ is the number $\left\lvert\vec{x}\right\rvert=\sqrt{\vec{x}\cdot\vec{x}}$.
13. The angle between non-zero vectors $\vec{x}$ and $\vec{y}$ is $\cos^{-1}\left(\frac{\vec{x}\cdot\vec{y}}{\left\lvert\vec{x}\right\rvert\left\lvert\vec{y}\right\rvert}\right)$.

Algorithms:

1. Suppose a given linear system has augmented matrix $M$. If the last column of $\mathrm{rref}(M)$ contains a pivot, then the system has no solutions. Otherwise, set the variables corresponding to columns of $\mathrm{rref}(M)$ that do not contain pivots equal to independent arbitrary parameters. Then the non-zero rows of $\mathrm{rref}(M)$ represent equations that express the remaining variables also in terms of these parameters. By letting the parameters range over all possible values, one obtains all solutions of the linear system.