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Sets
We will frequently work carefully with sets, so the material in this review section is very important.
Sets
We will frequently work carefully with sets, so the material in this review section is very important. If these topics are new to you, study this section carefully and consider consulting another text for a more comprehensive introduction.
Sets
Hard to get much more basic than that. Notice that the objects in a set can be anything, and there is no notion of order among the elements of the set. A set can be finite as well as infinite. A set can contain other sets as its objects. At a primitive level, a set is just a way to break up some class of objects into two groupings: those objects in the set, and those objects not in the set.
Example: Set membership
From the set of all possible symbols, construct the following set of three symbols, S&=\set{\blacksquare,\,\blacklozenge,\,\bigstar} Then the statement $\blacksquare\in S$ is true, while the statement $\blacktriangle\in S$ is false. However, then the statement $\blacktriangle\not\in S$ is true.
A portion of a set is known as a subset. Notice how the following definition uses an implication (if whenever then). Note too how the definition of a subset relies on the definition of a set through the idea of set membership.
If we want to disallow the possibility that $S$ is the same as $T$, we use the notation $S\subset T$ and we say that $S$ is a proper subset of $T$. We will do an example, but first we will define a special set.
Example: Subset
If $S=\set{\blacksquare,\,\blacklozenge,\,\bigstar}$, $T=\set{\bigstar,\,\blacklozenge}$, $R=\set{\blacktriangle,\,\bigstar}$, then T&\subseteq S & R&\not\subseteq T & \emptyset&\subseteq S \\ T&\subset S & S&\subseteq S & S&\not\subset S
What does it mean for two sets to be equal? They must be the same. Well, that explanation is not really too helpful, is it? How about: If $A\subseteq B$ and $B\subseteq A$, then $A$ equals $B$. This gives us something to work with, if $A$ is a subset of $B$, and vice versa, then they must really be the same set. We will now make the symbol $=$ do double-duty and extend its use to statements like $A=B$, where $A$ and $B$ are sets. Here is the definition, which we will reference often.
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Set Cardinality
On occasion, we will be interested in the number of elements in a finite set. Here is the definition and the associated notation.
Example: Cardinality and Size
If $S=\set{\blacklozenge,\,\bigstar,\,\blacksquare}$, then $\card{S}=3$.
Set Operations
In this subsection we define and illustrate the three most common basic ways to manipulate sets to create other sets. Since much of linear algebra is about sets, we will use these often.
Notice that the use of the word or in this definition is meant to be non-exclusive. That is, it allows for $x$ to be an element of both $S$ and $T$ and still qualify for membership in $S\cup T$.
Example: Set union
If $S=\set{\blacklozenge,\,\bigstar,\,\blacksquare}$ and $T=\set{\blacklozenge,\,\bigstar,\,\blacktriangle}$ then $S\cup T=\set{\blacklozenge,\,\bigstar,\,\blacksquare,\,\blacktriangle}$.
Example: Set intersection
If $S=\set{\blacklozenge,\,\bigstar,\,\blacksquare}$ and $T=\set{\blacklozenge,\,\bigstar,\,\blacktriangle}$ then $S\cap T=\set{\blacklozenge,\,\bigstar}$.
The union and intersection of sets are operations that begin with two sets and produce a third, new, set. Our final operation is the set complement, which we usually think of as an operation that takes a single set and creates a second, new, set. However, if you study the definition carefully, you will see that it needs to be computed relative to some universal set.
Notice that there is nothing at all special about the universal set. This is simply a term that suggests that $U$ contains all of the possible objects we are considering. Often this set will be clear from the context, and we will not think much about it, nor reference it in our notation. In other cases (rarely in our work in this course) the exact nature of the universal set must be made explicit, and reference to it will possibly be carried through in our choice of notation.
Example: Set complement
If $U=\set{\blacklozenge,\,\bigstar,\,\blacksquare,\,\blacktriangle}$ and $S=\set{\blacklozenge,\,\bigstar,\,\blacksquare}$ then $\setcomplement{S}=\set{\blacktriangle}$.
Condensed — the full section is in Beezer, A First Course in Linear Algebra.
Symbols used here
x belongs to A; every element of A is in B.
In either; in both; in A but not B.
Marks the point where the statement has been established.
A quantity with magnitude and direction; a column of numbers.
A rectangular array of numbers; a linear map.
Scaling factor of area/volume under A; zero means singular.
The matrix that undoes A; A with rows and columns swapped.
The factor by which an eigenvector is stretched: Av = λv.
Σ u_i v_i; the length of v, √(v·v).
Questions people ask
What does a determinant mean geometrically?
It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.
What is an eigenvector?
A direction the matrix does not turn — it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.
Why is matrix multiplication not commutative?
Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.
Sankhani wanu
Parts of this page are adapted from Beezer, A First Course in Linear Algebra (GFDL 1.2). Condensed and re-explained here; errors are ours.
Zambiri pa Linear Algebra
DeterminantsMatrix inverseEigenvalues and eigenvectorsMatrix multiplicationRow reductionVector spaces, span and linear independenceOrthogonality, projections and least squaresDiagonalisation and matrix powersLinear transformations and change of basis