Vector subspace facts for kids
A vector subspace is like a smaller vector space that lives inside a bigger one. Imagine a big room (the main vector space). A vector subspace is a smaller, special area within that room where you can still do all the same "vector space" activities, like adding things or multiplying them by numbers, and always stay within that special area.
Contents
What is a Vector Subspace?
A vector subspace is a special subset of a larger vector space. Think of it as a smaller collection of items that still follows all the rules of a vector space. For a set of items to be a vector subspace, it must be non-empty and meet three important conditions. These conditions make sure that the smaller set behaves just like a vector space on its own.
Key Rules for a Subspace
For a non-empty part W of a vector space V to be a vector subspace, it must follow these three rules:
Rule 1: The Zero Element
The "zero" item (also called the additive identity) must be part of W. This means that the starting point or origin, which represents nothing, must be included in your smaller space. If you're working with numbers, this would be the number 0. In vector spaces, it's the zero vector.
Rule 2: Staying Together (Addition)
If you take any two items from W and add them together, their sum must also be in W. This rule is called closed under addition. It means that adding items from your subspace won't take you outside of it. For example, if you have two vectors in your subspace, adding them together gives you another vector that is still in that same subspace.
Rule 3: Scaling Up (Multiplication)
If you take any item from W and multiply it by a regular number (called a scalar from a field like real numbers), the result must also be in W. This is known as closed under scalar multiplication. It means that stretching or shrinking an item from your subspace won't take you outside of it. For instance, if you have a vector in your subspace and you double its length, the new, longer vector is still within the same subspace.
Combining Subspaces
When you have two subspaces, let's call them W1 and W2, from the same main vector space V, you can combine them in certain ways. The sum of W1 and W2 (written as W1 + W2) is also a subspace. This sum includes all possible vectors you can get by adding a vector from W1 and a vector from W2. There's also something called the direct sum, which is a special kind of sum, and it also forms a subspace.
Related pages
