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)
A vector space is a set with two operations:
Vector addition: there is a vector
Scalar multiplication: there is a vector
These operations must satisfy the following properties:
VS3. such that ( is called a zero vector.)
VS4. such that ( is called an additive inverse of and is denoted by )
Theorem: Vector addition has the cancellation property. That is, if and then
By hypothesis, Applying VS1, we have Letting be the additive inverse of (using VS4), we have We use VS2 to regroup this as and VS4 to conclude that Finally, applying VS3 and VS1, we have as desired.☐
Theorem: The additive inverse of a vector is unique. In other words, if satisfy and then
It follows that The cancellation property implies ☐
We need to show that This follows from VS3.☐
Lecture 03 (2015-08-31)
We will temporarily leave the realm of abstract vector spaces and return to Recall that any vector is an -tuple of real numbers. We adopt the convention that be written when interpreted as a point, and when interpreted as a vector. Points are simply visualized as points in while vectors are visualized as directed line segments from the origin to the point
If then the distance from to is
This definition generalizes the Pythagorean theorem for
The length (a.k.a. norm, magnitude) of a vector is
i.e., the distance from to the origin.
How do scalar multiplication and vector addition interact with this geometric interpretation? If then
If then we say that the vectors and have the same direction. On the other hand, if then they have opposite directions.
We call two vectors parallel if one is a scalar multiple of the other, i.e., such that or
Under this definition, the zero vector is parallel to every vector.
If then can be interpreted as the fourth point of the parallelogram with legs and (situated at the origin). This can be confirmed by checking that the vector from to has the same slope as and vice versa. We call this interpretation the parallelogram law of vector addition.
If then their dot product is
This is an operation
Observe that This is the length formula.
Basic algebraic properties of the dot product (See Proposition 2.1 in Shifrin):