Lecture 1 Flashcards
Origin
The element of the set of real numbers R^n for every positive integer n whose components are all zero
When is an element zero?
If an element of R^n has a nonzero component
How do you obtain the sum of x+y?
By adding the corresponding components of x and y (component-wise addition)
Addition is commutative
x + y = y + x
Addition is associative
(x + y) + z = x + (y + z)
The opposite of x is defined and shown by
The opposite of x is -x and it is the element of R^n obtained by negating each component of x
What is the scalar multiple of x by s and how is it denoted?
Denoted as sx, it is the element of R^n obtained by multiplying each component of x by s.
When are two elements parallel?
Two elements x, y in R^n are parallel if x is a scalar multiple of y or if y is a scalar multiple of x.
When are two elements nonparallel?
When x nor y are a scalar multiple of the other
Define the dot product of x and y
x * y =x1y1+…+xnyn in R (it is important that it is in R and not R^n)
Define length (magnitude)
The length of x is the nonnegative real number |x| given by the square root of all components of x squared and added together
Define the Euclidean distance between x and y
Denoted by d(x, y), it is the nonnegative real number given by
d(x, y) = |x - y| = square root of (x1-y1)^2+…+(xn-yn)^2
What are the two ways to interpret elements of R^n
You can interpret them as either points/locations or vectors
How do you regard an element of R^n when it is a vector?
As a set of directions
How is an element (x1, x2,…,xn) of R^n regarded when it is a point?
It is interpreted as a point obtained by starting at the origin in R^n and traveling x1 units in one direction, x2 units in the other and so forth
