"Closed under linear combination" means that if you take any elements from a set and create a linear combination of them, the result will also be an element of that same set.
To break this down further:
- Set: A collection of objects (numbers, vectors, functions, etc.).
- Linear Combination: An expression formed by multiplying each element in a set by a scalar (a number) and then adding the results. For example, if we have elements v and w in a set, a linear combination would be av + bw, where a and b are scalars.
- Closed: A set is "closed" under an operation if performing that operation on elements of the set always produces another element that is also in the set.
Therefore, a set W is closed under linear combination if, for any vectors u and v in W and any scalars a and b, the vector *au + bv is also in W*. This property is fundamental to the concept of subspaces in linear algebra. It ensures that the subspace "contains" all possible linear combinations of its elements.
Here's a more formal definition using mathematical notation:
A set W is closed under linear combination if for all u, v ∈ W and for all scalars a, b ∈ F (where F is the field of scalars, commonly the real numbers or complex numbers), *au + bv ∈ W*.
Example:
Let's consider the set W of all vectors of the form (x, 0) in R2. This set represents the x-axis in the 2-dimensional plane. Is W closed under linear combination?
Let u = (x1, 0) and v = (x2, 0) be two vectors in W. Let a and b be any scalars. Then the linear combination *au + b*v is:
*au + bv = a(x1, 0) + b*(x2, 0) = (ax1 + bx2, 0)
Since the resulting vector has the form (something, 0), it also belongs to W. Therefore, the set W is closed under linear combination.
Why is this important?
The property of being "closed under linear combination" is crucial for defining subspaces. A subspace of a vector space must be closed under both addition and scalar multiplication (which are combined in the concept of linear combination). If a subset is closed under linear combination, it automatically satisfies the requirements to be a subspace (provided it's non-empty). Checking for closure under linear combination is a common way to verify if a given subset is a subspace.
In essence, "closed under linear combination" means a set behaves consistently under the fundamental operations of linear algebra: scaling and addition. If you start with elements in the set and apply these operations, you'll never "leave" the set.
