(24th-April-2021)
Scalars
: A scalar is just a single number, in contrast to most of the other
objects studied in linear algebra, which are usually arrays of multiple numbers.
We write scalars in italics. We usually give scalars lower-case variable names. When we introduce them, we specify what kind of number they are. For example, we might say “Let
s ∈ R
be the slope of the line,” while defining a real-valued scalar, or “Let
n∈ N
be the number of units,” while defining a natural number scalar.
Vectors
: A vector is an array of numbers. The numbers are arranged in order. We can identify each individual number by its index in that ordering.
Typically we give vectors lower case names written in bold typeface, such as x . The elements of the vector are identified by writing its name in italic
typeface, with a subscript. The first element of x is x 1 , the second element is x 2 and so on. We also need to say what kind of numbers are stored in the vector. If each element is in R, and the vector has n elements, then the vector lies in the set formed by taking the Cartesian product of R n times, denoted as R n .
When we need to explicitly identify the elements of a vector, we write them as a column enclosed in square brackets:
We can think of vectors as identifying points in space, with each element giving the coordinate along a different axis.
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