Math 260, Fall 2018, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 2.1.
2. Section 2.2.

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

1. Product (of two matrices).
2. Matrix form (of a linear system).
3. Linear (function, from $\mathbb{R}^m$ to $\mathbb{R}^n$).

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

1. Formula for the entries of the matrix product (this is Theorem 2.3.4 in the text).
2. Formula for $A\vec{x}$ in terms of the columns of $A$ and the entries of $\vec{x}$ (this is Theorem 1.3.8 in the text).
3. Theorem concerning commutativity of matrix multiplication (this is Theorem 2.3.3 in the text).
4. Theorem concerning associativity of matrix multiplication (this is Theorem 2.3.6 in the text).
5. Distributive properties of matrix multiplication (Theorem 2.3.7).
6. Theorem relating matrix multiplication and scalar multiplication (Theorem 2.3.8).
7. Theorem concerning the linearity of functions of the form $f(\vec{x})=A\vec{x}$.
8. Theorem concerning the representability by matrices of linear functions.

Solve the following problems:

1. Section 2.1, problems 1, 2, 3, 4, and 6.
2. Section 2.2, problems 2 and 5.
3. Section 2.3, problems 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, and 13.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Given an $n\times m$ matrix $A$ and an $m\times l$ matrix $B$, the product $AB$ is the $n\times l$ matrix whose $(i,j)$ entry is the dot product of the $i$th row of $A$ with the $j$th column of $B$.
2. Given a linear system $$\begin{align*}a_{1,1}x_1+\dots+a_{1,m}x_m&=b_1\\&\vdots\\a_{n,1}x_1+\dots+a_{n,m}x_m&=b_n,\end{align*}$$ put $$A=\begin{bmatrix}a_{1,1}&\dots&a_{1,m}\\\vdots&&\vdots\\a_{n,1}&\dots&a_{n,m}\end{bmatrix}\qquad\vec{x}=\begin{bmatrix}x_1\\\vdots\\x_m\end{bmatrix}\qquad\vec{b}=\begin{bmatrix}b_1\\\vdots\\b_n\end{bmatrix}.$$ Then the matrix form of the given system is the single matrix equation $A\vec{x}=\vec{b}$.
3. A function $f:\mathbb{R}^m\rightarrow\mathbb{R}^m$ is said to be linear if, for any $\vec{x},\vec{y}\in\mathbb{R}^m$ and any scalar $k$, we have (i) $f(\vec{x}+\vec{y})=f(\vec{x})+f(\vec{y})$ and (ii) $f(k\vec{x})=kf(\vec{x})$.

Theorems:

7. If $f$ has the form $f(\vec{x})=A\vec{x}$ for some matrix $A$, then $f$ is linear.

8. If $f$ is linear, then there exists a unique matrix $A$ such that $f(\vec{x})=A\vec{x}$. In fact, the $j$th column of $A$ is just $f(\vec{e}_j)$.