MATH 2500 Notes
Multivariable Calculus and Linear Algebra
As taught by Prof. Bruce Hughes
Vanderbilt University Fall Semester 2015
Transcribed by David K. Zhang

The following notes were taken live in-class during Prof. Hughes' MATH 2500 lectures. They are provided as a historical record of what topics were covered during each day of class. For readers who aim to learn this material for the first time, they are incomplete in several crucial ways:

  • Gaps in the logical flow of ideas due to missed classes.
  • Omitted discussions, proofs, and examples which I considered unnecessary.
  • Missing sections which were written down on paper and never typeset.

Lecture 01 (2015-08-26)

Introductory remarks and discussion of syllabus.

Lecture 02 (2015-08-28)

Definition: A vector space $V$ is a set with two operations:

  • Vector addition: $\forall \vx, \vy \in V,$ there is a vector $\vx + \vy \in V.$
  • Scalar multiplication: $\forall t \in \mathbb{R},$ $\forall \vx \in V,$ there is a vector $t\vx \in V.$

These operations must satisfy the following properties:

  • VS1. $\forall \vx, \vy \in V,$ $\vx + \vy = \vy + \vx.$
  • VS2. $\forall \vx, \vy, \vz \in V,$ $(\vx + \vy) + \vz = \vx + (\vy + \vz).$
  • VS3. $\exists \vo \in V$ such that $\forall \vx \in V,$ $\mathbf{0} + \vx = \vx.$ ($\vo$ is called a zero vector.)
  • VS4. $\forall \vx \in V,$ $\exists \vy \in V$ such that $\vx + \vy = \vo.$ ($\vy$ is called an additive inverse of $\vx,$ and is denoted by $-\vx.$)
  • VS5. $\forall s, t \in \mathbb{R},$ $\forall \vx \in V,$ $(st)\vx = s(t\vx).$
  • VS6. $\forall s \in \mathbb{R},$ $\forall \vx, \vy \in V,$ $s(\vx + \vy) = s\vx + s\vy.$
  • VS7. $\forall s, t \in \mathbb{R},$ $\forall \vx \in V,$ $(s + t)\vx = s\vx + t\vx.$
  • VS8. $\forall \vx \in V,$ $1\vx = \vx.$
Theorem: Vector addition has the cancellation property. That is, if $\vx, \vy, \vz \in V$ and $\vx + \vy = \vx + \vz,$ then $\vy = \vz.$
Proof: By hypothesis, $\vx + \vy = \vx + \vz.$ Applying VS1, we have $\vy + \vx = \vz + \vx.$ Letting $(-\vx)$ be the additive inverse of $\vx$ (using VS4), we have $(\vy + \vx) + (-\vx) = (\vz + \vx) + (-\vx).$ We use VS2 to regroup this as $\vy + (\vx + (-\vx)) = \vz + (\vx + (-\vx)),$ and VS4 to conclude that $\vy + \vo = \vz + \vo.$ Finally, applying VS3 and VS1, we have $\vy = \vz,$ as desired.
Theorem: The additive inverse of a vector is unique. In other words, if $\vx, \vy, \vz \in V$ satisfy $\vy + \vx = \vo$ and $\vz + \vx = \vo,$ then $\vy = \vz.$
Proof: It follows that $\vx + \vy = \vx + \vz.$ The cancellation property implies $\vy = \vz.$
Theorem: $-\vo = \vo.$
Proof: We need to show that $\vo + \vo = \vo.$ This follows from VS3.

Lecture 03 (2015-08-31)

We will temporarily leave the realm of abstract vector spaces and return to $\R^n.$ Recall that any vector $\vx \in \R^n$ is an $n$-tuple of real numbers. We adopt the convention that $\vx$ be written $(x_1, \ldots, x_n)$ when interpreted as a point, and $\mqty[x_1 \\ \vdots \\x_n]$ when interpreted as a vector. Points are simply visualized as points in $\R^n,$ while vectors are visualized as directed line segments from the origin $\vo$ to the point $\vx.$

Definition: If $\vx = (x_1, \ldots, x_n),\ \vy = (y_1, \ldots, y_n) \in \R^n,$ then the distance from $x$ to $y$ is
\[\norm{\vx - \vy} \coloneqq \sqrt{(x_1-y_1)^2 + \cdots + (x_n-y_n)^2}. \]

This definition generalizes the Pythagorean theorem for $n = 1,2,3.$

Definition: The length (a.k.a. norm, magnitude) of a vector $\vx = \mqty[x_1 \\ \vdots \\ x_n] \in \R^n$ is

\[\norm{x} \coloneqq \sqrt{x_1^2 + \cdots + x_n^2}, \]

i.e., the distance from $\vx$ to the origin.

How do scalar multiplication and vector addition interact with this geometric interpretation? If $c \in \R,$ $\vx \in \R^n,$ then

\[\norm{c\vx} = \sqrt{(cx_1)^2 + \cdots + (cx_n)^2} = \abs{c} \sqrt{x_1^2 + \cdots + x_n^2} = \abs{c} \norm{\vx}. \]
Definition: If $c > 0,$ then we say that the vectors $\vx$ and $c\vx$ have the same direction. On the other hand, if $c < 0,$ then they have opposite directions.
Definition: We call two vectors $\vx, \vy \in \R^n$ parallel if one is a scalar multiple of the other, i.e., $\exists c \in \R$ such that $\vy = c\vx$ or $\vx = c\vy.$

Under this definition, the zero vector $\vo$ is parallel to every vector.

If $\vx, \vy \in \R^2,$ then $\vx + \vy$ can be interpreted as the fourth point of the parallelogram with legs $\vx$ and $\vy$ (situated at the origin). This can be confirmed by checking that the vector from $\vy$ to $\vx + \vy$ has the same slope as $\vx,$ and vice versa. We call this interpretation the parallelogram law of vector addition.

Definition: If $\vx = \mqty[x_1 \\ \vdots \\ x_n], \vy = \mqty[y_1 \\ \vdots \\ y_n] \in \R^n,$ then their dot product is

\[\vx \cdot \vy \coloneqq x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]

This is an operation $\R^n \times \R^n \to \R.$

Observe that $\vx \cdot \vx = x_1^2 + \cdots + x_n^2 = \norm{\vx}^2.$ This is the length formula.

Basic algebraic properties of the dot product (See Proposition 2.1 in Shifrin):

  1. $\forall \vx, \vy \in \R^n,$ $\vx \cdot \vy = \vy \cdot \vx.$
  2. $\forall \vx \in \R^n,$ $\vx \cdot \vx \ge 0$ and $\vx \cdot \vx = 0$ iff $\vx = \vo.$
  3. $\forall c \in \R,$ $\forall \vx, \vy \in \R^n,$ $(c\vx) \cdot \vy = c(\vx \cdot \vy).$
  4. $\forall \vx, \vy, \vz \in \R^n,$ $\vx \cdot (\vy + \vz) = \vx \vdot \vy + \vx \cdot \vz.$

© David K. Zhang 2016 2021