Section 3.1: Definition and Examples (Vector Spaces) 1. Examples Euclidean Vector Spaces: The set of n-length vectors that we denoted by R n is a vector space. For simplicity, let s consider n = 2. A vector x = ( ) T x 1 x 2 in R 2 can be represented by a directed line segment from the origin to the point (x 1, x 2 ) or from any point (a, b) (same magnitude and direction). We scale vectors by any scalar α, multiplying each component by α. We add and subtract vectors via components, with a nice geometrical interpretation. The space of all m n matrices: Just as R n can be viewed as n 1 matrices, we can generalize to R m n, the set of all m n matrices. As we ve seen, matrices behave similarly to vectors in R n with addition/subtraction and scalar multiplication. Both of these vector spaces R n and R m n possess desirable operations that we can perform on their elements. These algebraic rules form the axioms used to define a general vector space. Intuitively, a vector space is a set that has the essential operations of addition and scalar multiplication defined, with some other well-desired properties satisfied. This concept, generalizing to defining abstract objects possessing certain properties, is one of the most powerful in all of mathematics.
M309 Notes, R.G. Lynch, Texas A&M Section 3.1: Definition and Examples (Vector Spaces) Page 2 of 6 Definition. An essential component of the definition are the closure properties: C1. If x V and α is a scalar, then αx V. C2. If x, y V, then x + y V. A vector space is closed under addition and scalar multiplication. Theorem. If V is a vector space and x is any element of V, then (i) 0x = 0 (ii) x + y = 0 implies that y = x (that is, the additive inverse is unique) (iii) ( 1)x = x Note. This is terrible notation and it is a shame that many authors introduce vector spaces as such. Addition and scalar multiplication are NOT(!!!) necessarily what you are used to in R n and this makes it look like it is. Usually, I will denote addition by and scalar multiplication by. While most vector spaces we will deal with seem to behave very similarly to R n, there are some weird ones (the last example). If I write + or, this will mean the usual addition and scalar multiplication. Example. When m and n are fixed, you can check that the R m n is a vector space. Is the set of all matrices (possibly different sizes) with the same addition and scalar multiplication a vector space?
M309 Notes, R.G. Lynch, Texas A&M Section 3.1: Definition and Examples (Vector Spaces) Page 3 of 6 Example (Lines in R 2 ). Let U := {(x, 1) : x R}, the set of all points lying on the horizontal line y = 1. Is U a vector space with usual addition and scalar multiplication? What about V := {(x, 0) : x R} with the same operations? What about W := {(x, mx) : x R} with the same operations?
M309 Notes, R.G. Lynch, Texas A&M Section 3.1: Definition and Examples (Vector Spaces) Page 4 of 6 Example (Continuous real-valued functions on [a, b]). Let C[a, b] := {f : [a, b] R : f is continuous} with addition and scalar multiplication defined by: (f g)(x) = f(x) + g(x) (α f)(x) = α f(x) Is C[a, b] a vector space? Example (Polynomials). Let P n [a, b] := {p : [a, b] R : p is a polynomial of degree less than n} with addition and scalar multiplication defined as on C[a, b]. Is P n [a, b] a vector space? Think about it. What happens if we require that the polynomials must be of degree exactly n?
M309 Notes, R.G. Lynch, Texas A&M Section 3.1: Definition and Examples (Vector Spaces) Page 5 of 6 Example (Shifted Reals). 1 Let S = {x : x R} be the set of real numbers with addition defined by and scalar multiplication defined by Is S a vector space? x y = x + y + 7 α x = α x + 7(α 1). 1 Dan Kalman s Notes, Example 2.
M309 Notes, R.G. Lynch, Texas A&M Section 3.1: Definition and Examples (Vector Spaces) Page 6 of 6 Example (R 2 Shifted in a Single Coordinate). 2 Let S be the set S = {x = (x 1, x 2 ) : x 1, x 2 R} scalar multiplication defined as usual, but with addition defined by x y = (x 1 + y 1, x 2 + y 2 + 1). Is S a vector space? 2 Thanks to Tom Vogel, also in the Texas A&M Math Department.