# Linear Spaces

## Definition: Linear Space

Let's begin by abstracting what we have done in the two cases above. The first is the easiest to think about. In the case of describing an object moving in three space, we have started with three objects for which we know some physical properties or have some intuitions about: the three basis directions. We then multiplied each direction be a length and added the resulting three vectors formally. We can abstract what we got as follows. A linear space (or a vector space) is a set of elements (vectors), V, and a set of numbers (scalars), S, (where for us, S will be either the real or complex numbers) satisfying the following properties: (Click here for a definition of the specialized math symbols used.)

• The set forms a group under addition. This means: it is closed, an identity exists, and each element has an inverse. Written out mathematically:
• If a, bV, then a + bV. (Note this means we have some definition of +).
• There exists a vector 0, such that for any aV, a + 0 = 0 + a = a.
• For every aV, ∃ (-a) ∈ V such that a + (-a) = (-a) + a = 0.
• The set forms a group under multiplication by a scalar. This means: it is closed, there is an identity, and every scalar has an inverse (except 0). Written out mathematically:
• If a V and α ∈ S, then αa V. (Note this means we have a definition of what it means to multiply a vector by a scalar).
• There exists a scalar 1 such that for any a V, 1⋅a=a⋅1 = a.
• For every &alpha: ∈ S (except for α = 0), ∃ α-1S, such that α-1a) = α(α-1a) = a.
• The two operations of addition and scalar multiplication are distributive. Mathematically, this means that addition and multiplication work like for regular numbers.
• ∀ α ∈ S, a, bV, α(a + b) = (αa) +(αb) and
∀ α, β ∈ S, aV, (α + β)a = (αa) +(βa)

These properties specify a linear (or vector) space. It is easy to show that the two examples discussed in the Motivation section both satisfy all these properties.

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