Vector spaces

• Main example. Spaces Rn.

• Definition of vector spaces.

• Other examples. Linear combinations.

• Examples of non-vector spaces.

• Vector subspaces.

Rn consists of all column vectors in n components:

x ∈ Rn, if x =

x1

x2

. . .xn

Two most important properties:1. For any x ∈ Rn and α ∈ R, there is a vector z = αx, and it is determined as

z1

z2

. . .zn

= α

x1

x2

. . .xn

=

αx1

αx2

. . .αxn

2. For any x, y ∈ Rn, there is a vector z = x + y, and it is determined as

z1

z2

. . .zn

=

x1

x2

. . .xn

+

y1

y2

. . .yn

=

x1 + y1

x2 + y2

. . .xn + yn

By construction the space Rn satisfies the definition of a vector space:A vector space V over a field R is a non-empty set V together with two operations+ (vector addition), and · (scalar multiplication) which satisfy the followingaxioms.

If a, b, c ∈ V, andα, β ∈ R, then

Closure under vector addition : a + b ∈ V (1)

Closure under scalar multiplication : αa ∈ V (2)

Commutativity : a + b = b + a (3)

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Associativity : (a + b) + c = a + (b + c) (4)

Existence of additive identity : ∃0 ∈ V : a + 0 = 0 + a = a (5)

Existence of additive inverses : ∃ − a ∈ V : a + (−a) = −a + a = 0 (6)

Distributive Law : α(a + b) = αa + αb (7)

Distributive Law : (α + β)a = αa + βa (8)

Existence of multiplicative identity : 1 · a = a (9)

Associativity of scalar multiplication : (α · β)a = α(βa) (10)

The definition of vector spaces is more general than just Rn. The followingexamples are also vector spaces. We separate them into two classes: Class A-spaces “equivalent” to Rn (this notion of equivalence will be discussed later),and Class B -spaces not “equivalent” to Rn.

Class A1. Space of polynomials of degree n− 1.2. Space of solutions of the ODE

dtx1 = a1x1,

dtx2 = a2x2,

. . .

dtxn = anxn.

3. Space of piece-wise linear approximations determined by the values at thegridpoints x1, x2, x3, . . . , xn.

Class B1. Spaces of sequences (x1, x2, . . . , xn, . . . ).2. Space of polynomials.3. Space of continuous functions.4. Space of analytic functions.5. Space of solutions of ∆u = 0 (without boundary conditions!).6. Space of functions with f(0) = 0.

Not every “space” is a vector space.Integers is not a vector space. And therefore space of ones money (with debt)is not a vector space.Positive numbers is not a vector space. And therefore space of distances on aline is not vector space.Space of (even oriented) distances between cities is not vector space, because ofthe triangle inequality.

Linear subspaces are (point-set) subspaces of linear spaces which are closedunder the two main properties of linear spaces - multiplication by a scalar andaddition.Example:Space of polynomials, space of piece-wise continuous functions, space of analytic

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functions etc. are also a linear subspaces of continuous functions.Multiplication by a scalar and addition is (often) combined into linear combi-nation property:For a, b ∈ V , and α, β ∈ R, c = αa + βb is their linear combination.

Questions: Is space of special matrices a linear space?1. All matrices2. n× n matrices3. n× n invertible matrices4. Upper/lower triangular matrices5. Permutation matrices

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