**Read** Sections 3.0 and 3.1 for next class.
Work through recommended homework questions.

**Tutorials:** No quizzes next week, focused on review.

**Help Centers:** Monday-Friday 2:30-6:30 in MC 106.

These **lecture notes** now available in pdf format as well,
a day or two after each lecture.
Be sure to let me know of technical problems.

**Theorem 2.4:** Write $A$ for the matrix with columns $\vv_1, \vv_2, \ldots, \vv_k$.
Then $\vv$ is a linear combination of $\vv_1, \vv_2, \ldots, \vv_k$ if and only if
the system with augmented matrix $[A \mid \vv \,]$ is consistent.

And we know how to determine whether a system is consistent! Use **row reduction**!

If $\span(S) = \R^n$, then $S$ is called a

**Example:** $\span(\ve_1, \ve_2, \ldots, \ve_n) = \R^n$.

**Example:** The span of $\vu = \colll 1 2 3$ and $\vv = \colll 4 5 6$
is the plane through the origin in $\R^3$
with direction vectors $\vu$ and $\vv$.

**Definition:** A set of vectors $\vv_1, \ldots, \vv_k$ is
**linearly dependent** if there are scalars $c_1, \ldots, c_k$,
__at least one of which is nonzero__, such that
$$
c_1 \vv_1 + \cdots + c_k \vv_k = \vec 0 .
$$
If the only solution to this system is the trivial solution
$c_1 = c_2 = \cdots = c_k = 0$,
then the set of vectors is said to be **linearly independent**.

Once again, this is something we know how to figure out! Use **row reduction**!

**Theorem 2.5**: The vectors $\vv_1, \ldots, \vv_k$ are linearly dependent
if and only if at least one of them can be expressed as a linear combination of the others.

**Fact:** Any set of vectors containing the zero vector is linearly dependent.

**Note:** You can sometimes see by inspection that some vectors are
linearly dependent, e.g. if they contain the zero vector, or if one
is a scalar multiple of another. Here's one other situation:

**Theorem 2.8:** If $m > n$, then any set of $m$ vectors in $\R^n$ is linearly
dependent.

**Theorem 2.7:**
Let $\vv_1, \vv_2, \ldots, \vv_m$ be row vectors in $\R^n$,
and let $A$ be the $m \times n$ matrix
$$ A = \collll {\vv_1} {\vv_2} {\vdots} {\vv_m}.$$
Then $\vv_1, \vv_2, \ldots, \vv_m$ are linearly dependent
if and only if $\rank(A) < m$.

We saw this by doing row reduction on $A$ and keeping track of how each new row is a linear combination of the previous rows. See Example 2.25 in the text.

**Questions?**

**Example 2.30:**
Consider a network of water pipes as in the figure to the right.

Some pipes have a known amount of water flowing (measured in litres per minute) and some have an unknown amount. Let's try to figure out the possible flows.

**Conservation of flow** tells us that the at each **node**,
the amount of water entering must equal the amount leaving.

Here are the constraints:

$$
\begin{aligned}
&\text{Node A}: & 5 + 10 &= f_1 + f_4 &\implies\quad f_1 + f_4 &= 15 \\
&\text{Node B}: & f_1 &= 10 + f_2 &\implies\quad f_1 - f_2 &= 10 \\
&\text{Node C}: & f_2 + f_3 + 5 &= 30 &\implies\quad f_2 + f_3 &= 25 \\
&\text{Node D}: & f_4 + 20 &= f_3 &\implies\quad f_3 - f_4 &= 20 \\
\end{aligned}
$$

The equations on the right have augmented matrix, which we row reduce:
$$
\bmat{rrrr|r}
1 & 0 & 0 & 1 & 15 \\
1 & -1 & 0 & 0 & 10 \\
0 & 1 & 1 & 0 & 25 \\
0 & 0 & 1 & -1 & 20
\emat
\longrightarrow
\bmat{rrrr|r}
1 & 0 & 0 & 1 & 15 \\
0 & 1 & 0 & 1 & 5 \\
0 & 0 & 1 & -1 & 20 \\
0 & 0 & 0 & 0 & 0
\emat
$$
The solutions are
$$
\begin{aligned}
f_1 &= 15 - t \\
f_2 &= \ph 5 - t \\
f_3 &= 20 + t \\
f_4 &= \phantom{20 + {}} t
\end{aligned}
$$
So if we control flow on AD branch, the others are determined.
In the text, flows are always assumed to be positive, so that
places constraints on $t$.

Because of $f_4$, we must have $t \geq 0$.

And from $f_2$, we must have $t \leq 5$.

The other constraints don't add anything, so we find that $0 \leq t \leq 5$.

This lets us determine the minimum and maximum flows: $$ \begin{aligned} 10 \leq& f_1 \leq 15 \\ 0 \leq& f_2 \leq 5 \\ 20 \leq& f_3 \leq 25 \\ 0 \leq& f_4 \leq 5 \end{aligned} $$

**Exercise 2.16:**
This figure represents traffic flow on a grid of one-way streets,
in vehicles per minute.

Since the same number of vehicles should enter and leave each intersection, we again get a system of equations that must be satisfied.

**On whiteboard:**

(a) set up and solve system

(b) if $f_4 = 10$, what are other flows?

(c) what are minimum and maximum flows on each street?

(extra) what can you say about how $f_2$ and $f_3$ compare?

(d) what happens if all directions are reversed?

(extra) what happens if the 5 changes to a 0 because of construction?

**Kirchhoff's Current Law** says that the sum of the currents flowing
into a node equals the sum of the currents leaving, just like for other networks.

We model devices in the circuit, such as light bulbs and motors, as
**resistors**, because they slow down the flow of current by taking
away some of the voltage:

**Ohm's Law**: voltage drop = resistance (in Ohms) times current (in amps):
$$ V = R I .$$
(The book uses $E$ for the voltage drop.)

**Kirchhoff's Voltage Law** says that the sum of the voltage drops
around a closed loop in a circuit is equal to the voltage provided by
the battery in that loop.

**On whiteboard:** did Exercise 2.4.20.

The other applications in Section 2.4, and the short Exploration on GPS after Section 2.4, are also quite interesting.

.