#subspaces

Finished Chapter 4: Vector Spaces.

Definition of Subspace A subspace is a set of a vectors with the property that linear combinations of these vectors remain in the set.

Geometrically, subspaces are represented by lines and planes that contain the origin.

Let V be a nonempty set of vectors in ℝⁿ. Then V is called a subspace if, whenever a and b are scalars and u and v are vectors in V, the linear combination au + bv is also in V.

Generally, a subspace contains the span of any finite set of vectors. In ℝⁿ, a subspace is exactly the span of finitely many of its vectors.

Theorem of Subspaces are Spans Let V be a nonempty set of vectors in ℝⁿ. Then V is a subspace of ℝⁿ if and only if there exists a set of vectors {u₁, ..., uk} in V that the following is true:

V = span{u₁, ..., uk}

Corollary of Subspaces are Spans of Independent Vectors If V is a subspace of ℝⁿ, then there exists a set of linearly independent vectors {u₁, ..., uk} in V, where V = span{u₁, ..., uk}.

Subspaces of ℝⁿ consist of spans of finite, linearly independent collections of vectors of ℝⁿ.

Definition of Basis of a Subspace Let V be a subspace in ℝⁿ. Then the vector set {u₁, ..., uk} is a basis for V is the following two conditions are satisfied:

1. V = span{u₁, ..., uk} 2. {u₁, ..., uk} is linearly independent.

The plural of basis is bases.

The main theorem about bases is not only they exist, but they must be of the same size.

Exchange Theorem Suppose the vector set {u₁, ..., uk} is linearly independent and each uk is contained in span{v₁, ..., vs}. Then s ≥ r.

Therefore, spanning sets of vectors have greater or equal amount of vectors as linearly independent sets do.

Theorem of Bases of ℝⁿ are of the Same Size Let V be a subspace of ℝⁿ and suppose the sets {u₁, ..., uk} and {v₁, ..., vm} are two bases for V. Then k = m.

Definition of Dimension of a Subspace Let V be a subspace of ℝⁿ. Then the dimension of V, written as dim(V), is defined as the number of vectors in a basis.

Corollary of Dimension of ℝⁿ The dimension of ℝⁿ is n.

Corollary of Linearly Independent and Spanning Sets of ℝⁿ The following properties are true in ℝⁿ:

From this discussion.

How do bots pull just the right item out of subspace? Each item put in there gets encoded as information, and that information is simply added to that bot’s conception of the item in their drives. They’re not even aware of it. They simply reach into their subspace for say, a spare missile, and their subspace system pulls the object they’re thinking of out of the pocket dimension, since it reads the code as part of the thought of that object. It’s an automatic thing.

Also, there are probably things it’s a bad idea to subspace. Like I think Dark Energon, with its chaotic nature, might run the risk of encoding wrong, or corrupting the data. It might just get lost, or cause other items to become lost. It might give you more than 99 of whatever item is in the 6th slot. Or you might explode?