Math 1600 Lecture 2

Math 1600 Lecture 2, Section 002, 9 Sept 2024

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Announcements:

Read Section 1.2 for next class. Work through suggested exercises.

Lecture notes (this page) available via link from Course Overview unit on Brightspace. Please read that unit if you haven't done so already.

Office hour: today, 3:30-4:00, MC130 (Middlesex College). Drop by with any questions! There will be more/longer office hours coming up, TBA. The Help Centre will probably start in a week, on Sept 16.

Homework 1 due Friday at 11:55pm. See course outline for flexibility in homework submission. Check that you can log into WeBWorK. Note that after registering it may take 48 hours before you can log in. [Do WeBWorK demo.]

We will finish Section 1.1 today, except that the part on Modular Arithmetic will be done later in the course.

Review of last lecture:

A vector can be represented by its list of components, e.g. $[1, 2, 3]$ is a vector in $\R^3$.
We write $\R^n$ for the set of all vectors with $n$ real components, e.g. $[1, 2, 3, 4, 5, 6, 7]$ is in $\R^7$.

We also often write vectors as column vectors, e.g. $\coll 1 2$.

Vector addition: $[u_1, \ldots, u_n] + [v_1, \ldots, v_n] := [u_1 + v_1, \ldots, u_n + v_n]$.
E.g. $[3, 2, 1] + [1, 0, -1] = [4, 2, 0]$.

Scalar multiplication: $c [u_1, \ldots, u_n] := [c u_1, \ldots, c u_n]$.

E.g. $2 [ 1 , 2, 3, 4, 5] = [2, 4, 6, 8, 10]$.

Zero vector: $\vec{0} := [0, 0, ..., 0]$.

New material: Section 1.1, continued: Properties of vector operations

Parallogram Law The picture to the right shows geometrically that vector addition is commutative: $\vec{u} + \vec{v} = \vec{v} + \vec{u}$.

In this true in $\R^n$? Let's check: $$ \begin{aligned} \vec{u} + \vec{v} &= \cyc{comm1}{ [u_1 + v_1,\, \ldots,\, u_n + v_n] }\\[2px] &= \cyc{comm2}{ [v_1 + u_1,\, \ldots,\, v_n + u_n] }\\[2px] &= \cyc{comm3}{ \vec{v} + \vec{u} .} \end{aligned} $$

Many other properties that hold for real numbers also hold for vectors:

Theorem 1.1: For vectors $\vu, \vv, \vw$ in $\R^n$ and scalars $c$ and $d$ in $\R$:
(a) $\ \vu + \vv = \vv + \vu$
(b) $\ (\vu + \vv) + \vw = \vu + (\vv + \vw)$
(c) $\ \vu + \vec{0} = \vu$
(d) $\ \vu + (-\vu) = \vec{0}$
(e) $\ c(\vu + \vv) = c\vu + c\vv$
(f) $\ (c + d)\vu = c\vu + d\vu$
(g) $\ c(d\vu) = (cd)\vu$
(h) $\ 1\vu = \vu$

These properties can be combined. For example, (a) and (c) tell us that $\vec{0} + \vu = \vu + \vec{0} = \vu$.

The properties can be used in both directions. E.g. (e) and (f) let you factor a common scalar or vector from a sum.

See the text for a proof of (b). The others are exercises that you should try.

Example: Simplification of an expression: $$ \begin{aligned} &3 \vec{b} + 2 (\vec{a} - 4 \vec{b})\\ =\ &\cyc{simp1}{3 \vec{b} + 2 \vec{a} - 8 \vec{b}}\\ =\ &\cyc{simp2}{2 \vec{a} - 5 \vec{b}} \end{aligned} $$

True/false: For every vector $\vu$, we have $2 \vu = \vu + \vu$.

True/false: For every vector $\vu$, we have $2 \vu \neq 3 \vu$.

Linear combinations

Definition: A vector $\vv$ is a linear combination of vectors $\vv_1, \vv_2, \ldots, \vv_k$ if there are scalars $c_1, c_2, \ldots, c_k$ so that \[ \vv = c_1 \vv_1 + \cdots + c_k \vv_k . \] The numbers $c_1, \ldots, c_k$ are called the coefficients. They are not necessarily unique.

Example: Is $\coll 1 {-1}$ a linear combination of $\coll 1 1$, $\coll 2 {-1}$ and $\coll 0 1$?

Example: Is $\coll 1 {-1}$ a linear combination of $\coll 1 0$ and $\coll 2 0$?

Example: Is $\coll 0 0$ a linear combination of $\coll 1 0$ and $\coll 2 0$?

Coordinates

Example: Express $\vw_1 = \coll 3 3$ as a linear combination of $\vu = \coll 2 1$ and $\vv = \coll {-1} 1$.

We can solve this by using $\vu$ and $\vv$ to make a new coordinate system in the plane. [Worked example: show that $\vw_1 = 2 \vu + \vv$.]

Similarly, show that $\vw_2 = \coll 4 {-1}$ can be expressed as $\vw_2 = \vu - 2 \vv$.

Note that in this case the coefficients are unique. In this situation, the coefficients are called the coordinates with respect to $\vu$ and $\vv$. So the coordinates of $\vw_1$ with respect to $\vu$ and $\vv$ are $2$ and $1$, and the coordinates of $\vw_2$ with respect to $\vu$ and $\vv$ are $1$ and $-2$.

Working in a different coordinate system is a powerful tool. coordinates example

An important real-world application of vectors: